Question

Find and classify all the critical points of the following functions. Hence sketch their graphs.

(a) y=3 x² +12 x-35
(b) f(x)=-2 x³ +15 x² -36 x+27

Answer 1

The critical points of the functions are

(a) y=3 x² +12 x-35

• x = -2

(b) f(x) = -2x³ + 15x² - 36x + 27

• x = 2 and x = 3

How to find the critical points

For the function y = 3x² + 12x - 35, to find critical points, compute the derivative y and set it to zero. Then solve for x

y' = 6x + 12

0 = 6x + 12

-6x = 12

x = -2

For f(x) = -2x³ + 15x² - 36x + 27,

y' = -6x² + 30x - 36

0 = -6x² + 30x - 36

solving the equation gives graphically gives

x = 2 and x = 3