Let f be a differentiable function defined on (0,2) such that f(x)0 and [f(x) + _0^x f(t)…

Question

Let $f$ be a differentiable function defined on $\left(0,\dfrac\pi2\right)$ such that $f(x)>0$ and

$$f(x) + \int_0^x f(t)\sqrt{1-(\log_e f(t))^2}\,dt = e, \quad \forall x\in\left(0,\frac\pi2\right)$$

Then $\left(6\log_e f\left(\dfrac\pi6\right)\right)^2$ is equal to \_\_\_

Answer: 27

Accepted Answer

Differentiating: $f'(x) + f(x)\sqrt{1-(\ln f(x))^2} = 0$. With $y=f(x)$:

$$\frac{dy}{dx} = -y\sqrt{1-(\ln y)^2} \implies \int\frac{dy}{y\sqrt{1-(\ln y)^2}} = -\int dx$$

Let $\ln y = t$: $\displaystyle\int\frac{dt}{\sqrt{1-t^2}} = -x+C \implies \sin^{-1}(\ln y) = -x+C$

At $x=0$, $f(0)=e \implies \ln f(0)=1 \implies \sin^{-1}(1) = C \implies C=\dfrac\pi2$.

$$\sin^{-1}(\ln f(x)) = \frac\pi2-x$$

At $x=\pi/6$: $\sin^{-1}(\ln f(\pi/6)) = \dfrac\pi3 \implies \ln f(\pi/6) = \sin(\pi/3) = \dfrac{\sqrt3}2$

$$\left(6\log_ef\left(\frac\pi6\right)\right)^2 = \left(6\cdot\frac{\sqrt3}2\right)^2 = (3\sqrt3)^2 = 27$$

Solutions · 1