Question

Which of the following statements is (are) TRUE?

• A. $\underset{x\to 0}{lim}\frac{|2x-1|-|2x+1|}{x}$ does not exist.
  
• B. The function $f\left(x\right)=\{\begin{array}{cc}2-x,& \text{if}x\le 1\\ x^2-3x+2,& \text{if}x>1\end{array}$ is differentiable at $x=1.$
• C. The absolute difference between the maximum and the minimum value of the function $f\left(x\right)=3{\sin }^{4}x-{\cos }^{6}x$ is $4$.
• D. The function $f\left(x\right)=x^5+10x^3+20x-18$ is strictly increasing $\forall x\in R$

Answer 1

Answer: (C), (D)

The function $f\left(x\right)=x^5+10x^3+20x-18$ is strictly increasing $\forall x\in R$

$f\left(x\right)=\{\begin{array}{cc}-\frac{2}{x},& \text{if}x\ge \frac{1}{2}\\ -4,& \text{if}-\frac{1}{2}<x<\frac{1}{2},x\ne 0\\ \frac{2}{x},& \text{if}x\le -\frac{1}{2}\end{array}$

Clearly, $\underset{x\to 0}{lim}f\left(x\right)$ exists and equals $-4$


$f\left(x\right)=\{\begin{array}{cc}2-x,& \text{if}x\le 1\\ x^2-3x+2,& \text{if}x>1\end{array}$

Checking continuity at $x=1$
$f\left(1^{-}\right)=1$
$f\left(1^{+}\right)=0$
$f\left(1^{-}\right)\ne f\left(1^{+}\right)$
So, $f$ is discontinuous at $x=1$
$\Rightarrow f$ is non-differentiable at $x=1$

$f\left(x\right)=3{\sin }^{4}x-{\cos }^{6}x$
$f^{\prime }\left(x\right)=12{\sin }^{3}x\cdot \cos x+6{\cos }^{5}x\cdot \sin x$
$=6\sin x\cos x(2{\sin }^{2}x+{\cos }^{4}x)$

$(Note:$ As $f\left(x\right)$ is periodic with fundamental period $\pi$, so it is sufficient to study range of $f\left(x\right)$ on the interval $[0,\pi ].)$
$f^{\prime }\left(x\right)=0\Rightarrow \sin 2x=0$
$\Rightarrow x=0,\frac{\pi }{2},\pi ,\dots$
$f\left(0\right)=-1,f\left(\frac{\pi }{2}\right)=3,f\left(\pi \right)=-1$
$\therefore$ Absolute difference between the maximum value and the minimum value $=4$

$f\left(x\right)=x^5+10x^3+20x-18$
$f^{\prime }\left(x\right)=5x^4+30x^2+20$
$f^{\prime }\left(x\right)>0\forall x\in R$
$\therefore f\left(x\right)$ is strictly increasing $\forall x\in R$