Question

If y = f(x) is differentiable at x = a and y = g(x) is continuous but not differentiable at x = a, g'(a⁺) & g'(a^–) are finite, then which one of the following is incorrect –

• A. y = f(x) g(x) is not differentiable at x = a, if f(1) ≠ 0
• B. y = f(x) g(x) is differentiable at x = a, if f(1) = 0
• C. y = f(x) g(x) is not differentiable at x = a, if f(1) = 0
• D. y = f(x) g(x) is continuous at x = a

Answer 1

Answer: (B)

y = f(x) g(x) is differentiable at x = a, if f(1) = 0

Answer 2

(f(a⁺) g(a⁺))′ = f ′(a⁺)g (a⁺) + f(a⁺)g ′ (a⁺)

= f ′(a⁺)g(1) + f(1) g ′ (a⁺)

(f(a^–)g(a^–))′ = f ′ (a^–) g(a^–) + f(a^–) g ′(a^–)

= f ′ (a^–) g (1) + f(1) g ′(a^–)

f(x) g(x) is differentiable if f(1) = 0

because f ′ (a⁺) = f ′ (a^–)