Question

Let $f\left(x\right)=\{\begin{array}{c}\frac{x^p}{(\sin x{)}^q},\text{if}0<x\le \frac{\pi }{2}\\ 0,\text{if}x=0\end{array},(p,q\in R)$ Then Lagrange's mean value theorem is applicable to$f\left(x\right)$ in closed interval $[0,x]$

• A. for all $p,q$
  
• B. only when $p>q$
• C. only when $p<q$
• D. for no value of $p,q$

Answer 1

The correct option is B only when $p>q$

Lagrange's mean value theorem is applicable to$f\left(x\right)$ in closed interval $[0,x]$ means the function is continuous in $[0,x]$
​​​​​​So,
$\underset{x\to 0^{+}}{lim}\frac{x^p}{(\sin x{)}^q}=0\underset{x\to 0^{+}}{lim}\frac{x^{p-q}}{\frac{(\sin x{)}^q}{x^q}}=0\underset{x\to 0^{+}}{lim}{x}^{p-q}=0\Rightarrow p>q$