How do you determine where the function is increasing or decreasing, and determine where relative maxima and minima occur for $f(x) = x^2e^x - 3$ ?

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How do you determine where the function is increasing or decreasing, and determine where relative maxima and minima occur for $f(x) = x^2e^x - 3$ ?

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Answer 1 · 2017-07-17T23:52:46

$f(x)$ is strictly increasing in $(-oo,-2)$ , has a local maximum in $x=-2$ , it is strictly decreasing in $(-2,0)$ reaching a local minimum in $x=0$ and again increasing for $x in (0,+oo)$

Explanation:

Given:

$f(x) = x^2e^x-3$

consider the derivative of the function:

$f'(x) = d/dx (x^2e^x-3) = (d/dx x^2)e^x + x^2 (d/dx e^x)$

$f'(x) = 2xe^x+x^2e^x$

$f'(x) = x(2+x)e^x$

Solve now the inequality:

$f'(x) > 0$

As $e^x > 0 AA x in RR$ , the sign of $f'(x)$ is the sign of $x(2+x)$ , then we have:

$f'(x) > 0$ for $x in (-oo,-2) uu (0,+oo)$

$f'(x) < 0$ for $x in (-2,0)$

and the critical points where $f'(x) = 0$ are $x_1 =-2$ and $x_2 =0$

We can conclude that $f(x)$ is strictly increasing in $(-oo,-2)$ , has a local maximum in $x=-2$ , it is strictly decreasing in $(-2,0)$ reaching a local minimum in $x=0$ and again increasing for $x in (0,+oo)$

graph{x^2e^x-3 [-10, 10, -5, 5]}

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How do you determine where the function is increasing or decreasing, and determine where relative maxima and minima occur for $f(x) = x^2e^x - 3$ ?

1 Answer

Explanation:

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Science

Math

Humanities

... and beyond

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 - 3f(x) = x^2e^x - 3?

1 Answer

Explanation:

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How do you determine where the function is increasing or decreasing, and determine where relative maxima and minima occur for $f(x) = x^2e^x - 3$ ?