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GATE 2026 CE (CE1) – Question 36

Engineering Mathematics · Calculus: Integrals, partial and total derivatives, gradient, divergence, curl, line, surface and volume integrals · 2 marks · Multiple choice

Let $f(x)$ be a continuous function defined in $[0, 2] \rightarrow \mathbb{R}$ and satisfying the equation $\int_0^2 f(x)[x - f(x)]\,dx = \frac{2}{3}$.

The value of $f(1)$ is

  1. 1
  2. 2
  3. $\frac{1}{2}$
  4. 0

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Correct answer: (C) $\frac{1}{2}$

Explanation

Complete the square: $f(x)[x - f(x)] = \frac{x^2}{4} - \left(f(x) - \frac{x}{2}\right)^2$. The integral of $\frac{x^2}{4}$ over $[0, 2]$ is $\frac{8}{12} = \frac{2}{3}$, which already equals the given value. So $\int_0^2\left(f - \frac{x}{2}\right)^2 dx = 0$, and since $f$ is continuous, $f(x) = \frac{x}{2}$. Then $f(1) = \frac{1}{2}$.