Question

Prove, using the second derivative, that the general quadratic y= ax^2+bx+c, is:

Answer 1

There are three things you have to know:

• A function is convex when its second derivative $f''(x)>0$
• A function is concave when it second derivative $f''(x)$
• The derivative of a power, $x^n$, is $nx^{n-1}$

So, the first derivative is

$y'=2ax+b$

and the second derivative is

$y''=2a$

This implies that the second derivative of a parabola is constant, and of course that 2 doesn't change the sign of $a$.