Answer:
The function is concave up on the interval (0, π/4]
And concave down on the interval [-π/4, 0)
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)