Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be a thrice differentiable odd function satisfying $f^{\prime}(\mathrm{x}) \geq 0, f^{\prime\prime}(\mathrm{x})=f(\mathrm{x}), f(0)=0, f^{\prime}(0)=3$. Then $9 f\left(\log _{\mathrm{e}} 3\right)$ is equal to ______.

JEE Main · 2025 · 02 Apr 2025 · Shift 1 · Differential Equations

Let

f : R \to R

be a thrice differentiable odd function satisfying

f^{'} (x) \geq 0, f^{''} (x) = f (x), f (0) = 0, f^{'} (0) = 3

. Then

9 f (lo g_{e} 3)

is equal to ______.

Answer: 36

The equation

f^{''} (x) = f (x)

has general solution

f (x) = A e^{x} + B e^{- x}

. Oddness forces

f (- x) = - f (x)

, hence

B = - A

and

f (x) = A (e^{x} - e^{- x}) = 2 A sinh x

, which also satisfies

f (0) = 0

. From

f^{'} (x) = 2 A cosh x

and

f^{'} (0) = 2 A = 3

A = \frac{3}{2}

, giving

f (x) = 3 sinh x

(with

f^{'} (x) = 3 cosh x \geq 0

as required). At

x = lo g_{e} 3

f (lo g_{e} 3) = 3 sinh (lo g_{e} 3) = 3 \cdot \frac{3 - \frac{1}{3}}{2} = 3 \cdot \frac{8/3}{2} = 4.

Therefore

9 f (lo g_{e} 3) = 36

JEE Main · 2025 · 02 Apr 2025 · Shift 1 · Integer answer

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