How do you use the first derivative test to determine the intervals on which $f(x)=x^4+3x^3+3x^2+1$ is increasing or decreasing and whether each critical point is a local maximum, minimum or neither?

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How do you use the first derivative test to determine the intervals on which $f(x)=x^4+3x^3+3x^2+1$ is increasing or decreasing and whether each critical point is a local maximum, minimum or neither?

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Answer 1 · 2015-09-24T23:05:39

See the explanation.

Explanation:

Using the rules:
$y=x^n, y'=nx^(n-1)$

and

$h(x)=f_1(x)+f_2(x)+...+f_n(x)$
$h'(x)=f'_1(x)+f'_2(x)+...+f'_n(x)$

we get:

$f'(x)=4x^3+9x^2+6x=x(4x^2+9x+6)$

Lets examine expression $4x^2+9x+6$ :
$4x^2+9x+6=0$
$D=b^2-4ac=9^2-4*4*6=81-96=-15$
$D<0$ which means that quadratic function $4x^2+9x+6$ doesn't have real zeros. Furthermore, $a=4>0$ (coefficient in front of the x^2), so the function is concave.

So, expression $4x^2+9x+6>0$ for $AAx in R$ .

$f'(x)=0 <=> x(4x^2+9x+6)=0 <=> x=0$

$AAx<0: f'(x)<0$ and $f$ is decreasing
$AAx>0: f'(x)>0$ and $f$ is increasing

$f'(x)$ changes sign in $x=0$ and $f(x)$ has minimum value $f_min=f(0)=1$

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How do you use the first derivative test to determine the intervals on which $f(x)=x^4+3x^3+3x^2+1$ is increasing or decreasing and whether each critical point is a local maximum, minimum or neither?

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Explanation:

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How do you use the first derivative test to determine the intervals on which f(x)=x^4+3x^3+3x^2+1f(x)=x^4+3x^3+3x^2+1 is increasing or decreasing and whether each critical point is a local maximum, minimum or neither?

1 Answer

Explanation:

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How do you use the first derivative test to determine the intervals on which $f(x)=x^4+3x^3+3x^2+1$ is increasing or decreasing and whether each critical point is a local maximum, minimum or neither?