Suppose you want to move from 0 to 100 on the number line. In each step, you either move right by a unit distance or you take a shortcut. A shortcut is simply a pre-specified
pair of integers i, j with i < j. Given a shortcut i, j if you are at position i on the number line, you may directly move to j. Suppose T(k) denotes the smallest number of steps needed to move from k to 100. Suppose further that there is at most 1 shortcut involving any number, and in particular from 9 there is a shortcut to 15. Let y and z be such that T(9) = 1 + min(T(y), T(z)). Then the value of the product yz is _____.

Consider the function f(x) = sin(x) in the interval x$\in $$\left[\mathrm{\pi}/4,7\mathrm{\pi}/4\right]$. The number and location(s) of the local minima of this function are

(A) One, at $\mathrm{\pi}$/2

(B) One, at 3$\mathrm{\pi}$/2

(C) Two, at $\mathrm{\pi}$/2 and 3$\mathrm{\pi}$/2

(D) Two, at $\mathrm{\pi}$/4 and 3$\mathrm{\pi}$/2

A point on a curve is said to be an extremum if it is a local minimum or a local maximum. The number of distinct extrema for the curve 3x^{4}- 16x^{3} + 24x^{2} + 37 is