Limit Continuity and Differentiability 7 Question 35
36. If $f(x)=\min {1, x^{2}, x^{3} }$, then
(2006, 3M)
(a) $f(x)$ is continuous everywhere
(b) $f(x)$ is continuous and differentiable everywhere
(c) $f(x)$ is not differentiable at two points
(d) $f(x)$ is not differentiable at one point
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Answer:
Correct Answer: 36. $1-\frac{2}{\pi}$
Solution:
- Given, $y=\frac{5 x}{3|1-x|}+\cos ^{2}(2 x+1)$
The function is not defined at $x=1$.
$$ \begin{aligned} & \Rightarrow \frac{d y}{d x}=\begin{array}{l} \frac{5}{3} \frac{(1-x)-x(-1)}{(1-x)^{2}}-2 \sin (4 x+2), \quad x<1 \\ \frac{5}{3} \frac{(x-1)-x(1)}{(x-1)^{2}}-2 \sin (4 x+2), \quad x>1 \end{array} \\ & \Rightarrow \quad \frac{d y}{d x}=\frac{\frac{5}{3(1-x)^{2}}-2 \sin (4 x+2), \quad x<1}{-\frac{5}{3(x-1)^{2}}-2 \sin (4 x+2), \quad x>1} \end{aligned} $$
