Solution assignment 07 Optimizing f(x)

Determine the extremes of the function:

$y=-x^3+9x^2-24x+26$

Verify in which point there is a maximum or a minimum by using the second derivative.

We find the candidates for extremes by calculating the first derivative and making it equal to $0$ :

$-3x^2+18x-24=0$

or:

$x^2-6x+8=0$

and this equation has the solutions:

$x=2$ or $x=4$

The second derivative is:

$y''=-6x+18$

For $x=2$ we have: $y''=6>0$ and thus there is a minimum for $x=2$

For $x=4$ we have: $y''=-6<0$ and thus there is a maximum for $x=4$ .