JEE Main · 2006 · 30 Apr 2006 · Offline · Application of Derivatives

WhatsApp Email

A triangular park is enclosed on two sides by a fence and on the third side by a straight river bank. The two sides having fence are of same length x. The maximum area enclosed by the park is

(A) $\frac{3}{2} x^{2}$
(B) $\frac{x ^{3}}{8}$
(C) $\frac{1}{2} x^{2}$ ✓ Correct
(D) $π x^{2}$

Answer: (C)

Solution

Let the two fenced sides, each of length

x

, meet at an angle

θ

, with the river forming the third side. The area of the triangle is

A = \frac{1}{2} x^{2} sin θ

, in which only

sin θ

varies. Since

sin θ

is largest, equal to

1

, when

θ = \frac{π}{2}

, the greatest area that can be enclosed is

A = \frac{1}{2} x^{2}

JEE Main · 2006 · 30 Apr 2006 · Offline · Single correct option

Practise more: all JEE Main PYQs on Application of Derivatives →

More JEE Main questions on Application of Derivatives

Let $f : R \to R$ be a twice differentiable function such that the quadratic equation $f (x) m^{2} - 2 f^{'} (x) m + f^{'} (x) = 0$ in m , has two equal roots for every …
JEE Main 2026View solution →
Let $f : R \to R$ be a twice differentiable function such that $f^{''} (x) > 0$ for all $x \in R$ and $f^{'} (a - 1) = 0$ , where a is real number. Let $g (x) = f (tan^{2} x - 2 tan x + a)$ , $0 < x < \frac{π}{2}$ .

Consider the following two statements:
(I) g i …
JEE Main 2026View solution →
Let the maximum value of $(sin^{- 1} x)^{2} + (cos^{- 1} x)^{2}$ for $x \in [- \frac{3}{2}, \frac{1}{2}]$ be $\frac{m}{n} π^{2}$ , where $gcd (m, n) = 1$ .

Then $m + n$ is equal to ____.
JEE Main 2026View solution →
Let $f (x) = x^{2025} - x^{2000}, x \in [0, 1]$ and the minimum value of the function $f (x)$ in the interval $[0, 1]$ be $(80)^{80} (n)^{- 81}$ . Then n is equal to
JEE Main 2026View solution →
Let $(2 α, α)$ be the largest interval in which the function $f (t) = \frac{∣ t + 1∣}{t ^{2}}, t < 0$ , is strictly decreasing. Then the local maximum value of the function
$g (x) = 2 lo g_{e} (x - 2) + α x^{2} + 4 x - α, x > 2$ , is ______
JEE Main 2026View solution →
Let f be a differentiable function satisfying
$f (x) = 1 - 2 x + \int_{0}^{x} e^{(x - t)} f (t) dt, x \in R$ and let …
JEE Main 2026View solution →