CMI-2018-DataScience-A: 14

Question

Consider the following functions defined from the interval $(0,1)$ to real numbers. Which of these functions attain their maximum value in the interval $(0,1)?$

• A. $f(x)=\frac{1}{x(1-x)}$
• B. $g(x)=-(x-0.75)^2$
• C. $u(x)=\sin(\frac{\pi x}{2})$
• D. $v(x)=x^2+2x$

Answer 1

for $g(x)=-(x-0.75)^2$ we have $g'(x)=-2(x-0.75)$

for extermum we $g’(x)$ will be 0.

$\therefore$ $g’(x)=0$

$\Rightarrow x=0.75$

Again $g’’(x)=-1<0$

$\therefore$ $x=0.75$ is a point of maximum