Question

Find the local maxima and local minima, if any of the following function. Also, find the local maximum and the local minimum values, as the case may be as follows.

$f\left(x\right)=x^3-6x^2+9x+15$

Answer 1

Given functions is, $f\left(x\right)=x^3-6x^2+9x+15$
$\Rightarrow f^{\prime }\left(x\right)=3x^2-12x+9andf"\left(x\right)=6x-12$
For maxima or minima put f'(x)=0
$\Rightarrow 3x^2-12x+9=0\Rightarrow 3(x^2-4x+3)=0\Rightarrow 3(x^2-3x-x+3)=0\Rightarrow 3\left[x\right(x-3)-1(x-3\left)\right]=0\Rightarrow 3(x-1)(x-3=0\Rightarrow x=1orx=3$
At $x=1f"\left(1\right)=6\times 1-12=6-12=-6<0$
$\therefore$ x=1 is a point of maxima.
Maximum value $=f\left(1\right)=(1{)}^3-6(1{)}^2+9\left(1\right)+15=1-6+9+15=19$
At $x=3,f"\left(3\right)=6\times 3-12=18-12=6>0$
x=3 is a point of minima
$\therefore$ Minimum value $=f\left(3\right)=3^3-6\times 3^2+9\times 3+15=27-54+27+15=15$