Is \(\displaystyle{f{{\left({x}\right)}}}={x}{e}^{{{x}}}-{x}^{{{2}}}\) concave or convex at

Trent Fuller

Trent Fuller

Answered question

2022-04-13

Is f(x)=xexx2 concave or convex at x=0?

Answer & Explanation

szalbierzfytg

szalbierzfytg

Beginner2022-04-14Added 13 answers

Step 1
You need to compute the second derivative in the point x=0, and check its sign. If it's positive, the function is convex, otherwise it's concave. So, we have
f(x)=xexx2
f(x)=ex+xex2x
fx)=ex+ex+xex2=ex(2+x)2
So, f(0)=e0(2+0)2=22=0
Step 2
Since the second derivative is negative immediatly before 0, and positive after, x=0 is an inflection point, in which the function passes from being concave to be convex, the scales of x and y axes are changed in order to emphatize the behaviour of the curve.

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