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:

  1. Given, $y=\frac{5 x}{3|1-x|}+\cos ^{2}(2 x+1)$

The function is not defined at $x=1$.

$$ \begin{aligned} & \Rightarrow \frac{d y}{d x}=\begin{array}{l} \frac{5}{3} \frac{(1-x)-x(-1)}{(1-x)^{2}}-2 \sin (4 x+2), \quad x<1 \\ \frac{5}{3} \frac{(x-1)-x(1)}{(x-1)^{2}}-2 \sin (4 x+2), \quad x>1 \end{array} \\ & \Rightarrow \quad \frac{d y}{d x}=\frac{\frac{5}{3(1-x)^{2}}-2 \sin (4 x+2), \quad x<1}{-\frac{5}{3(x-1)^{2}}-2 \sin (4 x+2), \quad x>1} \end{aligned} $$



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