Consider the function f(x)= x^2e^{-x}. Find the critical points of f.

Question:

Consider the function {eq}f(x)= x^2e^{-x}. {/eq} Find the critical points of f.

Critical Points:

{eq}\\ {/eq}

For a Function given by {eq}y=f(x) {/eq}, wwe can find the critical points by equating the first derivative to zero. We can also check for concavity and maxima/minima from the second derivative test with the help of critical points.

Answer and Explanation:

{eq}\\ {/eq}

Given : {eq}f(x)= x^2e^{-x}. {/eq}

For finding the critical points, we will differentiate the function and equate it to zero :-

{eq}\Rightarrow f'(x)=0\\\Rightarrow 2xe^{-x}-x^2e^{-x}=0\\\Rightarrow \dfrac{2x-x^2}{e^x}=0\\\Rightarrow \dfrac{x(2-x)}{e^x}=0\\\Rightarrow x=0,x=2 {/eq}

So, the critical points are {eq}x=0,2. {/eq}


Learn more about this topic:

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Finding Critical Points in Calculus: Function & Graph

from CAHSEE Math Exam: Tutoring Solution

Chapter 8 / Lesson 9
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