Find the intervals on which f is increasing and decreasing.

Question

Find the intervals on which f is increasing and decreasing.

f(x)= -2 sin^2 x on [-pi,pi]

Answer 1

A function is increasing when the first derivative is positive.

For f(x) = -2(sin(x))²

Using the chain rule:

f '(x) = -4sin(x)(cos(x)

Since we want a positive value, we need either sin(x) or cos(x) to be negative, but only one of them. On the interval of [-π, π] the 2nd quadrant has a negative cosine and positive sine, so that works. Also, the 4th quadrant has a negative sine and positive cosine, so that works. But at -π, 0, and π, we get zero for sine and at π/2 and -π/2, we get zero for cosine so the derivative is zero at those points (neither increasing or decreasing).

So the interval f is increasing is (-π/2, 0) and (π/2, π).

A function is decreasing when the first derivative is negative. That would occur when either both sine and cosine are positive or where both are negative. That would be in Q1 and Q3.

So the interval f is decreasing is (-π, -π/2) and (0, π/2).