Consider the functions f(x) = a sin 2x and g(x) = (sin x)/a, where a > 0 is a real number.

a. Graph the two functions on the interval [0, π/2], for a = 1/2, 1, and 2.

b. Show that the curves have an intersection point x^* (other than x = 0) on [0, π/2] that satisfies cos x^* = 1/(2a²), provided a > 1/√2.

c. Find the area of the region between the two curves on [0, x^*] when a = 1.

d. Show that as a →1/√2⁺. The area of the region between the two curves on [0, x^*] approaches zero.