Question

Let $f:[0,\frac{\pi }{2}]\to [0,1]$ be a differentiable function such that $f\left(0\right)=0,f\left(\frac{\pi }{2}\right)=1$, then

• A. $f^{{}^{\prime }}\left(\alpha \right)=\sqrt{1-\left(f\right(\alpha ){)}^2}$ for all $\alpha ϵ(0,\frac{\pi }{2})$
• B. $f^{{}^{\prime }}\left(\alpha \right)=\frac{2}{\pi }$ for all $\alpha ϵ(0,\frac{\pi }{2})$
• C. $f\left(\alpha \right)f^{{}^{\prime }}\left(\alpha \right)=\frac{1}{\pi }$ for at least one $\alpha ϵ(0,\frac{\pi }{2})$
• D. $f^{{}^{\prime }}\left(\alpha \right)=\frac{8\alpha }{{\pi }^2}$ for at least one $\alpha ϵ(0,\frac{\pi }{2})$

Answer 1

Answer: (C), (D)

The correct options are
C $f\left(\alpha \right)f^{{}^{\prime }}\left(\alpha \right)=\frac{1}{\pi }$ for at least one $\alpha ϵ(0,\frac{\pi }{2})$
D $f^{{}^{\prime }}\left(\alpha \right)=\frac{8\alpha }{{\pi }^2}$ for at least one $\alpha ϵ(0,\frac{\pi }{2})$

Let : $[0,\frac{\pi }{2}]\to [0,1]$ be a ........
(A) Consider $g\left(x\right)=si{n}^{-1}f\left(x\right)-x$
Since $g\left(0\right)=0,g\left(\frac{\pi }{2}\right)=0$
$\therefore$ There is at least one value of $\alpha ϵ(0,\frac{\pi }{2})$ such that
$g^{{}^{\prime }}\left(\alpha \right)=\frac{f^{{}^{\prime }}\left(\alpha \right)}{\sqrt{1-\left(f\right(\alpha ){)}^2}}-1=0$
i.e. $f^{{}^{\prime }}\left(\alpha \right)=\sqrt{1-\left(f\right(\alpha ){)}^2}$ for atleast one value of $\alpha$ but may not be for all $\alpha ϵ(0,\frac{\pi }{2})$
$\therefore$ false

(B) Consider $g\left(x\right)=f\left(x\right)-\frac{2x}{\pi }$
Since $g\left(0\right)=0,g\left(\frac{\pi }{2}\right)=0$
$\therefore$ there is at least one value of $\alpha ϵ(0,\frac{\pi }{2})$ such that
$g^{{}^{\prime }}\left(\alpha \right)=f^{{}^{\prime }}\left(\alpha \right)-\frac{\pi }{2}=0$
i.e. $f^{{}^{\prime }}\left(\alpha \right)=\frac{2}{\pi }$ for atleast one value of $\alpha$ but may not be for all $\alpha ϵ(0,\frac{\pi }{2})$
$\therefore$ false

(C) Consider $g\left(x\right)=\left(f\right(x){)}^2-\frac{2x}{\pi }$
Since $g\left(0\right)=0,g\left(\frac{\pi }{2}\right)=0$
$\therefore$ there is at least one value of $\alpha ϵ(0,\frac{\pi }{2})$ such that
$g^{{}^{\prime }}\left(\alpha \right)=2f^{(}\alpha \left)f^{{}^{\prime }}\right(\alpha )-\frac{\pi }{2}=0$
$\therefore f^{(}\alpha \left)f^{{}^{\prime }}\right(\alpha )=\frac{1}{\pi }$
$\therefore$ True

(D) Consider $g\left(x\right)=f\left(x\right)-\frac{4x^2}{{\pi }^2}$
Since $g\left(0\right)=0,g\left(\frac{\pi }{2}\right)=0$
$\therefore$ there is at least one value of $\alpha ϵ(0,\frac{\pi }{2})$ such that
$g^{{}^{\prime }}\left(\alpha \right)=f^{{}^{\prime }}\left(\alpha \right)-\frac{8\alpha }{{\pi }^2}=0$
$\therefore f^{{}^{\prime }}\left(\alpha \right)=\frac{8\alpha }{{\pi }^2}$
$\therefore$ True