Question

1. Use the definition of convexity to prove that the function f(x) = x2 - 4x + 8 is convex. Is
this function strictly convex?
2. Use the definition of convexity to prove that the function f(x)= ax + b is both convex and
concave for any a•b ≠ 0.

Answer 1

Defination : A function is said to be convex if for all   ,   satisfies   for all .

1.  Given function ,   .

We know that ,   for all real number x and y .

for all .

Hence the function is convex .

2. Given function , .

Now ,

So , both and   holds true and hence the function is both convex and concave .

.

.

.

If you have any doubt please comment .