A function f is defined on the interval [0, π]. Its derivative is given by f'(x) = cos x−sin 2x (...)

Question

a) At witch interval is the function f is increasing?

b) Estimate to two decimal places the location of a point x > 0 where f has a point of inflection.

Answer 1

First, find the critial points by solving the equation f'(x) = 0.

 

cosx - sin(2x) = 0

 

cosx - 2sinxcosx = 0

 

cosx(1 - 2sinx) = 0

 

cosx = 0 or sinx = 1/2

 

cosx = 0 when x = π/2

 

sinx = 1/2 when x = π/6 or 5π/6

 

Evaluate f'(x) at a point in each of the following intervals:  (0,π/6), (π/6, π/2), (π/2, 5/6), (5π/6, π)

 

On (0, π/6), f'(x) > 0.  So f is increasing.

 

On (π/6, π/2), f'(x) < 0.  So f is decreasing.

 

On (π/2, 5π/6), f'(x) > 0.  So, f is increasing

 

On (5π/6, π), f'(x) < 0.  So, f is decreasing.

 

Answer 2

a)

 

Set f'(x) = 0    and solve for x.

 

cos(x) - sin(2x) = 0

 

cos(x) - 2sin(x)cos(x) = 0

 

cos(x)(1 - 2sin(x)) = 0

 

cos(x)(1 - 2√(1 - cos²(x)) = 0

 

 

Then use first derivative test.  The derivative must positive when evaluating test points.

 

 

b)

 

Set the f"(x) equal to zero and solve for x.

 

-sin(x) - 2cos(2x) = 0

 

-sin(x) - 2(1 - sin²x) = 0

 

sin²(x) - sin(x) - 2 = 0

 

 

Then use second derivative test.  The derivative must change signs in order to be a point of inflection.