ISI2014-DCG-21

Question

Suppose that the function $h(x)$ is defined as $h(x)=g(f(x))$ where $g(x)$ is monotone increasing, $f(x)$ is concave, and $g’’(x)$ and $f’’(x)$ exist for all $x$. Then $h(x)$ is

• A. always concave
• B. always convex
• C. not necessarily concave
• D. None of these

Answer 1

The correct answer is option C.

Consider f(x) is a function that is twice continuously differentiable on an interval I. Then the function f(x) is 

1. convex if $f''(x)>0 ,$ for all x in I.
2. concave if $f''(x)<0 ,$for all x in I.

let take ,

$f(x)=-x^{2}$  

This graph is a concave graph as 

$f''(x)=-2<0$

Figure,

now ,

Let’s take $g(x)=e^{x}$ ,It is monotone increasing function.

Figure,

Now if we see $h(x)=g(f(x))=e^{-x^{2}}$

$h''(x)=2e^{-x^{2}}\left [ 2x^{2}-1 \right ]$.

It is concave when $-\frac{1}{\sqrt{2}}<x<\frac{1}{\sqrt{2}}$ range

it is convex when $x>\frac{1}{\sqrt{2}}$ and $x<-\frac{1}{\sqrt{2}}$

Graph ,

As concaveness or convexnees we can find out by second differentiation . 

As it is clear that it will never always concave or convex so option a and b is false.