Question

Given f(x) = x²e^{2(x – 1)}, 0 £ x £ 1= a sgn (x + 1) cos (2x – 2) + b x², 1 < x £ 2, f(x) is differentiable at x = 1 provided

• A. a = – 1, b = 2
• B. a = 1, b = – 2
• C. a = – 3, b = 4
• D. a = 3, b = – 4

Answer 1

Answer: (A)

a = – 1, b = 2

Answer 2

f(1^–) = 1 = f(1) and f(1⁺) = a + b

For continuity at x = 1, a + b =1 ……(i)

In 0 < x < 1, f '(x) = 2xe^{2(x – 1)} + 2x²e^{2(x – 1)}

\ f '(1^–) = 4

In 0 < x < 2, f ' (x) = – 2a sin (2x – 2) + 2bx

\ f '(1⁺) = 2b

For differentiability at x = 1, 2b = 4. This with (1) gives the values a = – 1, b = 2.