Question

Determine the intervals on which the function is concave up or concave down. (Enter your answers using interval notation. If an answer does not exist, enter DNE.)
f(θ) = 15 θ + 15 sin2(θ), [0, π]

Answer 1

Answer:

Step-by-step explanation:

attached is the solution

Answer 2

Answer:

The function is concave up on the interval (0, π/4]

And concave down on the interval [-π/4, 0)

Step-by-step explanation:

To investigate if a function is concave up or concave down, we investigate the second derivative of the function.

Given f(θ) = 15θ + 15sin²θ

Let us differentiate this function twice in succession.

f'(θ) = 15 + 15(2sinθcosθ) = 15 + 15sin2θ

f''(θ) = 30sin2θ.

The function is concave upward when it's second derivative is greater than zero. That is, when

f''(θ) > 0

=> 30sin2θ > 0

=> sin2θ > 0

=> 0 < θ ≤ π/4

The interval is (0, π/4]

The function is concave down when it's second derivative is less than zero. That is when

f''(θ) < 0

=> 30sin2θ < 0

=> sin2θ < 0

=> -π/4 ≤ θ < 0

The interval is [-π/4, 0)