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Prove, using the second derivative, that the general quadratic y= ax^2+bx+c, is:
a) always convex when a>0
b) always concave when a <0

Respuesta :

There are three things you have to know:

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

So, the first derivative is

[tex]y'=2ax+b[/tex]

and the second derivative is

[tex]y''=2a[/tex]

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

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