Mathematics

Let $$f(x)\, =\, 3x^{2}.\, \sin \,\displaystyle \frac{1}{x}\, -\, x\cos\, \displaystyle \frac{1}{x},\, x\, \neq\, 0, f(0)\, =\, 0\, f \left (\, \displaystyle \frac{1}{\pi} \right )\, =\, 0$$, then which of the  following is/are not correct.


ANSWER

$$f(x)$$ is non-differentiable at $$x = 0$$

$$f(x)$$ is discontinuous at $$x = 0$$

$$f(x)$$ is differentiable at $$x = 0$$


SOLUTION
$$f(x)\, =\, 3x^{2}. \sin \displaystyle \frac{1}{x}\, -\, x.\cos\, \displaystyle \frac{1}{x}$$
$$\Rightarrow\, f(x)\, =\,\displaystyle  \int \left ( 3x^{2}.\sin \displaystyle \frac{1}{x}\, -\, \cos\, \displaystyle \frac{1}{x} \right ) dx$$
$$=\, x^{3}\, \sin\, \displaystyle \frac{1}{x}.\, -\, \int \cos\, \frac{1}{x}\, \left (\, -\, \displaystyle \frac{1}{x^{2}} \right )\, x^{3}\, dx\, -\, \int x \cos \displaystyle \frac{1}{x}dx$$
$$=\, x^{3}\, \sin \, \displaystyle \frac{1}{x}\, +\, C$$ 
Since $$f \left ( \displaystyle \frac{1}{\pi} \right )\, =\, 0\, +\, C \, \Rightarrow\, C\, =\, 0$$
$$\Rightarrow\, f(x)\left \{\begin{matrix} x^{3}\sin\, \displaystyle \frac{a}{x} & , & x \, \neq\, 0 \\  0 & , & x = 0 \end{matrix} \right \}$$

$$F(x)$$ is clearly continuous and differentiable at $$x = 0$$ zero with $$f`(0) = 0$$.

$$\displaystyle f\, (0)\, =\, \lim_{h \rightarrow 0} \frac{3h^2\, \sin \frac{1}{h}\, -\, h\, \cos \frac{1}{h}}{h}$$ $$\displaystyle =\, 3h\sin\frac{1}{h}\, -\, \cos \frac{1}{h}$$

This limit doesn't exist, hence $$f(x)$$ is non-differentiable at $$x = 0$$.

Also $$\displaystyle \lim_{x \rightarrow 0}\, f`(x)\, =\, 0$$. 

Thus $$f`(x)$$ is continuous at $$x = 0$$.

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Multiple Correct Hard Published on 17th 09, 2020
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