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*

Let f, g : R → R+ defined by f (x) = 2x + 3 and g(x) = x2. Find (f ◦ g)(x) and (g ◦ f )(x)

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.

Define g : R → R byg(x) =(x2 cos(1/x) if x̸ = 0,0 if x = 0.Show that g is differentiable at x for every x ∈ R

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.

Fie functiile f, g : R −→ R, definite prin f (x) = x − 2, g(x) = 2 − x, ∀x ∈ R(a) Sa se calculeze (f ◦ g)(1)