) Let f, g be differentiable functions on an interval I. Suppose that a, b ∈ Iwith f (b) ≥ g(b) and f ′(x) ≤ g′(x) on [a, b]. Show that f (x) ≥ g(x) on[a, b].

Let f, g : R → R be given functions. Suppose that f and g are continuous at c ∈ R.Prove that the functionl(x) := inf{f (x), g(x)}, x ∈ R,is continuous at c.

Let f(x) and g (x) be one-to-one functions, and let h (x) = g ( f(x)). Show that the inverse functionof h (x) is h^−1 (x) = f^ −1(g^−1 (x))

Let [a, b] be a bounded closed interval, f ∈ R[a, b] and M > 0. Suppose that |f (x)| ≤ Mfor all x ∈ [a, b]. Prove thatZ baf ≤ M (b − a)

Problem 2. Prove that the function f : R → R, f (x) = |x|3 is twice differentiable at anypoint a ∈ R, but is not three-times differentiable at 0.

Let f and g be two twice differentiable functions defined form R → R. If f (x) has a local maximum at x = c and g(x) has a local minimum at x = c, then*