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, the which one of the following is INCORRECT ?

• A. y = f(x) g(x) is not differentiable at x = a, if f(1)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 '(1) g(1) + f(1) g'(a^–)

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

because f '(a^–) = f '(a⁺)