Question

Examine the function,
f(x)=2x^3- 9x^2- 60x+150
for extreme values.

Answer 1

$y=2x^3-9x^2-60x+150\\ y'=6x^2-18x-60\\ 6x^2-18x-60=0\\ x^2-3x-10=0$

According to Viet's theorem,

$x_1+x_2=3\\ x_1x_2=-10\\ x_1=5\\ x_2=-2$

$x\in(-\infty,-2), y'>0,\\ x\in(-2,5), y'<0,\\ x\in(5,\infty), y'>0.$

Then

$x=-2$ is a local maximum, $y_{max}=218$ ;

$x=5$ is a local minimum, $y_{min}=-125$ .