Verify that the function satisfies the three hypotheses of Rolle's Theorem on the given interval. Then find all numbers c that satisfy the conclusion of Rolle's Theorem. f(x)=sin(x/2), interval [(pi/2), (3pi/2)

The function f is defined by f(x) = x² - 2 - 4 sin (3x² + 5). Let l be the line tangent to the graph of f at x = 1. Find all values of x in the interval [0, 2] where the line tangent to the graph of f is parallel to , excluding x = 1. You may use a calculator and round to the nearest thousandth.

Find the intervals where the function concaves up and down for f(x) = 8 sin(x) + cot(x),    −π ≤ x ≤ π

Use The Intermediate Value Theorem to prove that there is at least one solution to tan(0) = (0 - 1)² on the interval (-7,7). (Note, tan() is undefined for = ±, so you will need to find your own a and b to use in the IVT)