Question

If f is positive and concave upward on an interval I, what is the concavity of the function g(x) = [f(x)]^2 on the interval I? Justify your claim.

I am having troubles doing this question if it is squared wouldn't it all be concave down?

Thanks.

Answer 1

\[f(x)=x^2\] fits that bill

Answer 2

and \[f^2(x)=x^4\] is concave up for sure

Answer 3

\[g(x)=[f(x)]^2\implies g'(x)=2f(x)f'(x)\implies g''(x)=2[f'(x)]^2+2f(x)f''(x)\]
Given that \(f\) is positive and concave up in some interval \(I\), you know that \(f(x)>0\) and \(f''(x)>0\). What sign must \(g''(x)\) have?

Answer 4

It must be positive?

Answer 5

Right, so if the second derivative is positive, what do you know about the concavity of \(g\)?

Answer 6

It is concave up!

Answer 7

Precisely

Answer 8

Thanks!

Answer 9

yw