Consider the following function. h(x) = sin2(x) + cos(x) 0 < x < 2𝜋

Question

Consider the following function.

h(x) = sin²(x) + cos(x)    0 < x < 2𝜋

Find h'(x).

h'(x) =

Excluding any values not in the domain of h, for what values of c is h'(c) = 0

 or undefined? (Enter your answers as a comma-separated list.)

c=

Give the critical numbers of the function. (Enter your answers as a comma-separated list.)

x=

Answer 1

h(x) = sin²x + cosx

h'(x) = 2sinxcosx - sinx = sinx(2cosx - 1)

h'(x) = 0 when sinx = 0 or cosx = 1/2

x = π, π/3, 5π/3

Answer 2

Sometimes it's easier to think about sin²(x) as [sin(x)]². Both expressions are the same but using [sin(x)]² you can see the application of the chain rule. So the derivative, using first the power rule, then the chain rule, is 2[sin(x)]¹•(sin(x))' = 2sin(x)cos(x)

The derivative of the second term, cos(x), is -sin(x)

So h'(x) = 2sin(x)cos(x) - sin(x)

For the second question, for what values of c is h'(c) = 0:

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

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

So either sin(x) = 0 (meaning x = π) or 2cos(x) - 1 = 0 (meaning cos(x) = 1/2, so x = π/3 or 5π/3)

So c = π/3, π, and 5π/3

These values are the critical numbers on the domain specified.