Question

The area bounded by the curve y = ƒ(x), y = x and the lines x = 1, x = t is (t + ) – $\sqrt { 2 }$ – 1 sq. units, for all t > 1. If ƒ(x) satisfying ƒ(x) > x for all x > 1, then ƒ(x) is equal to-

• A. x + 1 +
• B. x +
• C. 1 +
• D.

Answer 1

Answer: (A)

x + 1 +

Answer 2

It is given that ƒ(x) > x, for all x > 1. So, Area bounded by y = ƒ(x), y = x and the lines x = 1, x = t is given by

But this area is given equal to (t + – $\sqrt { 2 }$– 1) sq. units. Therefore,

dx = t + – $\sqrt { 2 }$ – 1, for all t > 1 On

differentiating both sides w.r.t. t, we get

ƒ(t) – t = 1 + for all t > 1

Ž ƒ(t) = t + 1 + for all t > 1

Hence ƒ(x) = x + 1 + for all x > 1.

Hence (1) is the correct answer.