Find the critical numbers of f

Question

F(x) = 5sin(x)cos(x) on [-pi,pi]

1.Find the critical numbers of f

2.Determine the intervals on which f is increasing and decreasing

I found the derivative but I'm not sure where to go from here

f'(x)= cos^2(x) - sin^2(x)

Answer 1

f(x) = 5/2sin2x; f'(x) = 5cos2x = 0; cos2x = 0; 2x = ±π/2; ±3π/2; x = ±π/4, ±3π/4;

Critical numbers: -3π/4, -π/4, π/4, 3π/4.

(-π, -3π/4), (-π/4, π/4), (3π/4, π) are intervals of increasing;

(-3π/4, -π/4), (π/4, 3π/4) are intervals of decreasing

Answer 2

Critical points for f occur at any x-values for which f^' = 0 or is undefined.

You got f^' correct except it should be 5 times that (which won't affect the location of f's critical points).

f^'(x) = 5[cos²x - sin²x] = 5cos2x = 0 (This is a double-angle identity.)

cos2x = 0

x = - 3π/4 , - π/4 , π/4 , 3π/4

f is increasing on (- π , -3π/4) ∪ (- π/4 , π/4) ∪ (3π/4 , π) and decreasing elsewhere.