True or False Questions - Calculus

Question

State, with explanation, if the following statements are True or False.

1. If f’(5) = 1.8 and g’(5) = 4.2 then f(x) + g(x) has slope 6 at x = 5
2. It is possible for y = f(x) to have an inflection point at (a, f(a)) even if f’(a) is not defined
3. For all functions f, if f’(x) exists ∀x, then f “(x) exists ∀x

Answer 1

(1) True. Differentiation is linear, so (f+g)'(x) = f'(x) + g'(x). Hence, (f+g)'(5) = f'(5) +g'(5) = 1.8 + 4.2 = 6.

(2) True. Take, for instance, f(x) = x^1/3. Then (0, 0) is an inflection point, but f'(0) is undefined.

(3) False. Consider f(x) = (3/4) * x^4/3. Then f'(x) = x^1/3 is defined everywhere, but f''(x) = (1/3) * x^-2/3 is undefined at x=0.