Classifying Critical Points and Extreme Values for a Function

Key Questions

How do you find and classify the critical points of $f(x)=x^3$ ?

Here is how to find and classify a critical point of $f$ .

Remember that $x=c$ is called a critical value of $f$ if $f'(c)=0$ or $f'(c)$ is undefined.

$f'(x)=3x^2=0 Rightarrow x=0$ is a critical number.

(Note: $f'$ is defined everywhere, $0$ is the only critical value.)

Observing that $f'(x)=3x^2 ge 0$ for all $x$ ,

$f'$ does not change sign around the critical value $0$ .

Hence, $f(0)$ is neither a local maximum nor a local minimum by First Derivative Test.

Wataru · · Sep 28 2014

Questions

How do you find and classify the critical points of $f(x)=x^3$ ?
How do you find the critical points of a rational function?
How do you know how many critical points a function has?
How many critical points can a cubic function have?
How many critical points can a function have?
How many critical points can a quadratic polynomial function have?
What is the first step to finding the critical points of a function?
How do you find the absolute extreme values of a function on an interval?
How do you find the extreme values of the function and where they occur?
What is the extreme value of a quadratic function?
How do you calculate $f '(x)$ and use calculus to find the maximum value of $sin (ln x)$ on the interval [1, 10]?
What is the local maximum and or miminum of the function $f(x)=2x^2+2x^2-4x$ ?
How do you use the closed-interval method to find the absolute maximum and minimum values of the function $f(x)=x-2sinx$ on the interval $[-π/4, π/2]$ ?
How do you find the absolute maximal and minimal values of $f(x)=x^3−6x^2+9x+4$ on the interval $[−1, 5]$ ?
What is the overall maximum value of the function $f(x)=(18x)/(4+x^2)$ on the interval (0,4)?
How do you find all local maximum and minimum for $g(x)=6 x^3−(144 )x^2 +(1080 )x−4$ ?
How do you find the relative maxima and minima of the function $f(x)= 7(x^2 - 16)^2$ ?
How do you find the max and min for $f(x)= x - ((64x)/(x+4))$ on the interval [0,13]?
What are the maximum and minimum values that the function $f(x)=x/(1 + x^2)$ ?
How do you determine the maximum and minimum value of the function $f(x)=-xe^x + 2$ ?
How to find critical points and the highest point on the graph of $4x^2 + 4xy+7y^2=15$ ?
How do you find the exact minimum value of $f(x) = e^x + e^(-2x)$ on [0,1]?
How do you use the Extreme Value Theorem to determine the global extrema of the function $f(x)=5+ (x^2)/(x+2)$ on the closed interval [-1,3]?
Question #15c3e
What are the values and types of the critical points, if any, of $f(x) = abs(x^2-1)$ ?
What are the values and types of the critical points, if any, of $f(x) =4x^3+48x^2+26x-246$ ?
What are the values and types of the critical points, if any, of $f(x) =x^3 + 3x^2-24x$ ?
What are the values and types of the critical points, if any, of $f(x)=300/(1+.03x^2)$ ?
What are the values and types of the critical points, if any, of $f(x)=xlnx$ ?
What are the values and types of the critical points, if any, of $f(x)=x^2-5x+4$ ?
What are the values and types of the critical points, if any, of $f(x)=x / (x^2 + 4)$ ?
What are the values and types of the critical points, if any, of $f(x)=x^3-6x^2+12x-6$ ?
What are the values and types of the critical points, if any, of $f(x)=cos^2 x - sin^2 x$ ?
What are the values and types of the critical points, if any, of $f(x,y)=2x + 4y + 4$ ?
What are the values and types of the critical points, if any, of $f(x,y)=y^2-x^2+3x-sqrt(2xy-4x)$ ?
What are the values and types of the critical points, if any, of $f(x)=x^3 + 3x^2 + 1$ ?
What are the values and types of the critical points, if any, of $f(x)=x^4 - 8x^3 + 18x^2 - 27$ ?
What are the values and types of the critical points, if any, of $f(x,z) = x^4 + 15z^2 + 2xz^2 - 456z^2$ ?
What are the values and types of the critical points, if any, of $f(x,y) = x(2-x-y)$ ?
What are the values and types of the critical points, if any, of $f(x,y) = sqrty(5-2x^2-3y ))$ ?
What are the values and types of the critical points, if any, of $f(x)=7x^4-6x^2+1$ ?
What are the values and types of the critical points, if any, of $f(x)=x^4/(6x^2+x)$ ?
What are the values and types of the critical points, if any, of $f(x)=3e^(-2x^2))$ ?
What are the values and types of the critical points, if any, of $f(x)=e^(-2x^2)-xe^x$ ?
What are the values and types of the critical points, if any, of $f(x)=(2x^2+5x+5)/(x+1)$ ?
What are the values and types of the critical points, if any, of $f(x) = (x^4/5)(x-4)^2$ ?
What are the values and types of the critical points, if any, of $f(x) = (x-5)/(x-3)^2$ ?
What are the values and types of the critical points, if any, of $f(x) =x^2-sqrtx$ ?
What are the values and types of the critical points, if any, of $f(x,y)=xy(1-8x-7y)$ ?
What is the difference between a critical point and a stationary point?
What are the values and types of the critical points, if any, of $f(x)=(x+3x^2) / (1-x^2)$ ?
What are the values and types of the critical points, if any, of $f(x)=2x^5(x-3)^4$ ?
What are the values and types of the critical points, if any, of $f(x,y)=4y(1 - x^2)$ ?
What are the values and types of the critical points, if any, of $f(x, y) = x^3+y^3-3*x*y-7$ ?
What are the values and types of the critical points, if any, of $f(x, y) = xy^2+xy-3x^2y-7x$ ?
Is it possible to determine the critical points of a function without using the function's derivatives?
How do you find the maximum and minimum values of the function $f(x)= x - ((64x)/(x+4))$ on the interval [0,13]?
How do you find the relative extrema for $f(x) = x^2(6-x)^3$ ?
How do you find the relative extrema for $f(x) = (x)3^-x$ ?
How do you find the relative extrema for $f(x) = x - log_4x$ ?
How do you find the relative extrema for $f(x) = (x^2 - 3x - 4)/(x-2)$ ?
How do you find the relative extrema for $f(x) = (x+2)/x$ ?
How do you find the relative extrema for $f(x)=x^3-4x^2+x+6$ ?
How do you find the relative extrema for $f(x) =2x- 3x^(2/3) +2$ ?
How do you find the relative extrema for $f(x)=(x^4)+(4x^3)-12$ ?
How do you determine where the function is increasing or decreasing, and determine where relative maxima and minima occur for $2x^3 + 3x^2 - 12x$ ?
How do you determine where the function is increasing or decreasing, and determine where relative maxima and minima occur for $f(x)=x^3-3x^2+1$ ?
How do you determine where the function is increasing or decreasing, and determine where relative maxima and minima occur for $f(x)=x^3+5x^2$ ?
How do you determine where the function is increasing or decreasing, and determine where relative maxima and minima occur for $f(x) = 3x^4 + 16x^3 + 24x^2 + 32$ ?
How do you determine where the function is increasing or decreasing, and determine where relative maxima and minima occur for $y=cos^2 x - sin^2 x$ ?
How do you determine where the function is increasing or decreasing, and determine where relative maxima and minima occur for $f (x) = x + 4/x$ ?
How do you determine where the function is increasing or decreasing, and determine where relative maxima and minima occur for $f(x) = sqrt(x^2+1)$ ?
How do you determine where the function is increasing or decreasing, and determine where relative maxima and minima occur for $f(x)=(x^3)/(x^2-4)$ ?
How do you determine where the function is increasing or decreasing, and determine where relative maxima and minima occur for $f (x) = x^3 + 6x^2$ ?
How do you determine where the function is increasing or decreasing, and determine where relative maxima and minima occur for $f(x)= 3x^4+4x^3-12x^2+5$ ?
How do you determine where the function is increasing or decreasing, and determine where relative maxima and minima occur for $F(x)= (x^2)(lnx)$ ?
How do you determine where the function is increasing or decreasing, and determine where relative maxima and minima occur for $f(x) = x² + 2x - 3$ ?
How do you determine where the function is increasing or decreasing, and determine where relative maxima and minima occur for $y = x^3 - 3x^2 - 9x +15$ ?
How do you determine where the function is increasing or decreasing, and determine where relative maxima and minima occur for $f(x) = (x - 1)/x$ ?
How do you determine where the function is increasing or decreasing, and determine where relative maxima and minima occur for $f(x) = x^2e^x - 3$ ?
How do you determine where the function is increasing or decreasing, and determine where relative maxima and minima occur for $f(x) = 2x + ln x$ ?
How do you determine where the function is increasing or decreasing, and determine where relative maxima and minima occur for $f(x)= 2x+3/x$ ?
How do you determine where the function is increasing or decreasing, and determine where relative maxima and minima occur for $y=x^4-2x^3$ ?
How do you determine where the function is increasing or decreasing, and determine where relative maxima and minima occur for $f(x) = x ln x$ ?
How do you determine where the function is increasing or decreasing, and determine where relative maxima and minima occur for $f(t) = 6t + 1/t$ ?
How do you determine where the function is increasing or decreasing, and determine where relative maxima and minima occur for $3x^5 - 5x^3$ ?
How do you find the local max and min for $f(x) = (3x) / (x² - 1)$ ?
How do you find the local max and min for $f(x) = 2 x + 3 x ^{ -1 }$ ?
How do you find the local max and min for $f(x) = x / (x^2 + 81)$ ?
How do you find the local max and min for $y = 3x^4 + 4x^3 – 12x^2 + 1$ ?
How do you find the local max and min for $2x^3-3x^2-36x-3$ ?
How do you find the local max and min for $f(x)=-sinx-cosx$ ?
How do you find the local max and min for $f(x) = 2x^3 - 5x +1$ in the interval is (-3,3)?
How do you find the local max and min for $f(x)=x^3-2x+5$ on the interval (-2,2)?
How do you find the local max and min for $f (x) = x^(3) - 6x^(2) + 5$ ?
How do you find the local max and min for $f(x) = e^x(x^2+2x+1)$ ?
How do you find the local max and min for $f(x) = x^4 - 2x^2 + 1$ ?
How do you find the local max and min for $y=2x^3 - 5x^2 - 4x + 7$ ?
How do you find the local max and min for $x^5 ln x$ ?
How do you find the local max and min for $f(x)= 5x - 3$ ?
How do you find the local max and min for $f(x)= x^2 + 8x -12$ ?
How do you find the local max and min for $f(x)= sinx$ ?
How do you find the local max and min for $f(x) = 7x + 9x^(-1)$ ?
How do you find the local max and min for $f(x) = x^3 - 27x$ ?
How do you find the local max and min for $f(x) = 1 - sqrt(x)$ ?
How do you find the local max and min for $f(x)=2x^3 + 5x^2 - 4x - 3$ ?
How do you find the local max and min for $x^3-2x^2-8x$ ?
How do you find the local max and min for $f(x)= (x^2)/(x-2)^2$ ?
How do you find the local max and min for $F(x) = ln((x^4) + 27)$ ?
How do you find the local max and min for $f(x)=x+(1/x)$ ?
How do you find the local max and min for $y = x^3 − 12x + 9$ ?
Question #da337
What is the minimum value of the function $y=sqrt(x²+2ax+10a²)$ where a > 0 ?
Given $f'(x) = (x+1)(x-2)²g(x)$ where $g$ is a continuous function and $g(x) < 0$ for all $x$ . On what interval(s) is $f$ decreasing?
How do you determine the absolute extreme values for the function $y=x(sqrt(1-x²))$ on its domain?
How do you find the absolute and local extreme values for $y=-x+7$ on the interval [-10,10]?
How do you find the absolute and local extreme values for $f(x)=(x-3)^2-9$ on the interval [-8,-3]?
How do you find the critical points for $f(x)=-x^2+6x+2$ ?
How do you find the critical points for $f(x)=x^3-2x^2+3x$ ?
How do you find the critical points for $y=x^4-3x^3+5$ ?
How do you find the critical points for $g(x)=2x^3-3x^2-12x+5$ ?
How do you find the critical points for $y=x-sqrtx$ ?
How do you find the critical points and local max and min for $y=4x-x^2$ ?
How do you find the critical points and local max and min for $y=(x-1)^4$ ?
How do you find the critical points and local max and min for $g(x)=2x^3-24x+5$ ?
How do you find the critical points to sketch the graph $f(x)=7+6x-x^2$ ?
How do you find the critical points to sketch the graph $g(x)=x^4-8x^2-10$ ?
How do you find the critical points to sketch the graph $y=x(x+2)^2$ ?
How do you find the critical points to sketch the graph $h(x)=27x-x^3$ ?
Question #2b61c
Question #e5c2c
How do you find the relative extrema of $f(x)=3x^5-5x^3$ ?
How do you find the maxima and minima of the function $f(x)=x^3+3x^2-24x+3$ ?
Question #be380
Question #c8903
Question #cbfb8
Question #2237b
Consider the function $y = e^x$ defined on $[-1, 2]$ . What are the maximum and minimum of the function on this interval?
Question #a621d

Graphing with the First Derivative

View all chapters

Interpreting the Sign of the First Derivative (Increasing and Decreasing Functions)
Identifying Stationary Points (Critical Points) for a Function
Identifying Turning Points (Local Extrema) for a Function
Classifying Critical Points and Extreme Values for a Function
Mean Value Theorem for Continuous Functions

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Classifying Critical Points and Extreme Values for a Function

Key Questions

Questions

Graphing with the First Derivative