Question

Show that the sum of two concave functions is concave. Is the product of two concave functions also concave?

Answer 1

Answer with explanation:

Let us assume that the 2 functions are:

1) f(x)

2) g(x)

Now by definition of concave function we have the first derivative of the function should be strictly decreasing thus for the above 2 function we conclude that

$\frac{d}{dx}\cdot f(x)$

Now the sum of the 2 functions is shown below

$y=f(x)+g(x)$

Diffrentiating both sides with respect to 'x' we get

$\frac{dy}{dx}=\frac{d}{dx}\cdot f(x)+\frac{d}{dx}\cdot g(x)$

Since each term in the right of the above equation is negative thus we conclude that their sum is also negative thus

$\frac{dy}{dx}$

Thus the sum of the 2 functions is also a concave function.

Part 2)

The product of the 2 functions is shown below

$h=f(x)\cdot g(x)$

Diffrentiating both sides with respect to 'x' we get

$h'=\frac{d}{dx}\cdot (f(x)\cdot g(x))\\\\h'=g(x)f'(x)+f(x)g'(x)$

Now we can see the sign of the terms on the right hand side depend on the signs of the function's themselves hence we remain inconclusive about the sign of the product as a whole. Thus the product can be concave or convex.