Analyze function g ( θ ) = 1 θ + 5 ( sin ⁡ ( θ ) )^ 2 on the interval [0,𝜋]

Question

Analyze function g(θ)=1θ+5(sin⁡(θ))^2 on the interval [0,𝜋]

a) Use interval notation to indicate where g is concave up:

b) Interval where g is concave down:

c) Infection points of g in interval [0,pi]:

Answer 1

a) Find where second derivative is positive:

b) Find where second derivative is negative.

c) Find where second derivative is 0 and the 1st derivative sign doesn't change around that point ( If it did, than it is a likely min or max)

g(θ) = θ + 5sin²(θ)

g' = 1 + 10sin(θ)cos(θ)

g'' = 10(cos²(θ) -sin²(θ))

setting g'' = 0 yields cos²θ = sin²θ which happens at pi/4 and 3pi/4

You should be able to work out the questions exploring the signs of the 2nd dervative.