Let g: RR be a differentiable function satisfying the following conditions. • g(0) = 1 and g(t) ≥ 0 for all t = R. • The derivative function g' : R → R is continuous. Argue that the following inequality holds: 2 g(t) dt- t = [["9 (t)³ dt| ≤

Give an example of a differentiable function ƒ whose first derivative is zero at some point c even though ƒ has neither a local maximum nor a local minimum at c.

Is it true that every function y = ƒ(x) that is differentiable on [a, b] is itself the derivative of some function on [a, b] ? Give reasons for your answer.

Let f be a function on [0, 7] whose second order derivative is continuous and f(T) = 1. Suppose that | (f(2) + f"(x)) sin a da = 4. Compute f(0).

Determine whether the following set of functions are linearly independent f(x) = k1 , g(x) = sin? (k2x) ,h(x) = cos²(k,x)

Consider the following function which is defined for all x and y: f(x, y) = 2(1-p²)x²y² - x² - y² + 3pxy + 2x+4y+2 where p is a constant. (a) Find the first order derivatives of f and enter them as functions of x, y and p (b) Find the second order derivatives of f and enter them as functions of x, y and p (c) For p = 1, find the stationary point (x*, y*).

Let f and g be functions satisfying that f(1) = 3, g(1) = 4, ƒ'(1) = 1, g' (1) = 5. For the following function h(x), find h' (1). Answer: h(x) = 2ƒ(x) + 3g(x).