Determine rather strange density by accept-reject algorithm

Question

Solve the following problem:

The rather strange density

f(x) ∝ exp(-x²/2) {sin(6x)² + 3 cos(x)² sin(4x)² +1}

can be generated using the Accept-Reject algorithm.

a. Plot f(x) and show that it can be bounded by Mg(x), where g is the standard normal density

g(x) = exp(-x²/2) / √2π

Find an acceptable if not necessarily optimal value of M.

b. Generate 2500 random variables from f using the Accept-Reject algorithm.

c. Deduce from the acceptance rate of this algorithm an approximation of the normalizing constant of f, and compare the histogram with the plot of the normalized f.