Limits and Deri...

Which of the following is not true about $$y=f(x)$$?

Differential coefficient of $$\sec (\tan^{-1} x)$$ w.r.t. x is

Given that $$f (x)$$
is a differentiable function of $$ x$$ and that $$f(x)$$ . $$f (y)$$ =  $$f (x) $$+ $$f (y)$$ + $$f (xy) -2$$ and that
$$f (2) =5$$.

Then $$f (3)$$ is equal to?

lf $${f}'({x})={g}({x})$$ and $${g}'({x})=-{f}({x})$$ for all $$x$$ and $${f}(2)=4= {g}(2)$$, then $${f}^{2}(24)+{g}^{2}(24)$$ is

Assertion(A): Let $${ f }({ x })$$ be twice differentiable function such that $$f^{ '' }(x)=-{ f }({ x })$$ and $$f^{ ' }(x)={ g }({ x })$$. lf $${ h }({ x })=[{ f }({ x })]^{ 2 }+[{ g }({ x })]^{ 2 }$$ and $${ h }(1)=8$$, then $${ h }(2)=8$$

Assertion (A): lf $$f(x)=\cos^{2}x+\cos^{2}\left(x+\dfrac{\pi}3\right)- \cos x \cos \left(x+\dfrac{\pi}3\right)$$ then $$f'(x)=0$$

lf $$ \displaystyle \mathrm{f}(\mathrm{x})=\mathrm{x}.\sin \frac{1}{x}$$ for $$x \neq 0,\ \mathrm{f}(\mathrm{0})=0$$ then?

The graph of the function $$y = f (x)$$ has a unique tangent at the point $$(e^{a} ,0)$$ through which the graph passes then $$\displaystyle \lim_{x\rightarrow e^{a}}\frac{log_{e}\{1+7f(x)\}-sinf(x)}{3f(x)}$$ is

$$f(x)=|x-1|+|x-3|$$ then $$f^{'}(2)=$$

 Let $$\mathrm{f}(\mathrm{x})=\left\{\begin{array}{l}\mathrm{x}^{\mathrm{n}}\sin\frac{1}{\mathrm{x}},\quad  \mathrm{x}\neq 0\\0, \quad \mathrm{x}=0\end{array}\right.$$ , then f(x) is continuous but not differentiable at x=0 if