How to Count Critical Points Given the Derivative? (calculator allowed)

Question

Consider the function h(x) whose derivative is given by,

h'(x) = (cos²(x)/2x) - (1/12)

How many critical points does h(x) have on the open interval (0, 10)?

A) 3

B) 4

C) 5

D) 6

Answer 1

The critical points are where the first derivative = 0 or undefined.

Graph on Desmos: 0<x<10 it has 3 solutions, 3x = 0. 3 critical points.

Not sure on a TI 84 family if you have to add 1/12 to both sides: cos^2(x)/2x = 1/12

Cross multiply: 12cos^2(x) = 2x

Divide by 2.

6 cos^2(x) = x

Graph y = x and y = 6cos^2(x). You'll see they cross 3 times (same x values where h'(x) has zeroes.

A