Continuity and ...

If $$H(x_0)=0$$ for some $$x=x_0$$ and $$\dfrac {d}{dx}H(x) > 2cxH(x)$$ for all $$x \le x_0$$, where $$c > 0$$, then

Given a function $$f:[0,4] \rightarrow \mathrm{R}$$ is differentiable, then for $$\displaystyle \operatorname{some} \alpha, \beta \in(0,2), \int_{0}^{4} f(t) d t$$ equals to

$$ f(x)=x^{2}+x g^{\prime}(1)+g^{\prime \prime}(2) $$ and $$ g(x)=f(1) x^{2}+x f^{\prime}(x)+f^{\prime \prime}(x) $$
The value of $$ f(3) $$ is

If f(x) = $$\displaystyle \int_{0}^{x}(f(t))^2 dt f:R→R$$ be differentiable function and f(g(x)) is differentiable function at x=a, then

$$y=f(x)$$ is

If $$f:R \rightarrow (0, \infty)$$ be a differentiable function $$f(x)$$ satisfying $$f(x+ y) - f(x - y) = f(x) \{ f(y) - f(y) - y \}, \forall x, y \epsilon R, (f(y) \neq f(-y)$$ for all $$y \epsilon R)$$ and $$f' (0) = 2010$$.
Now answer the following questions

Which of the following is true for $$f(x)$$

If $$u=\sin^{-1}\Bigg(\dfrac{2x}{1+x^2}\Bigg)$$ and $$v=\tan^{-1}\Bigg(\dfrac{2x}{1-x^2}\Bigg)$$, then $$\dfrac{du}{dv}$$ is

The function $$f(x)=\Bigg\{\dfrac{\sin x}{x}+\cos x,if\,x\neq 0\\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,k,if\,x=0$$  is continuous at $$x=0$$,then the value of k is

The function $$f(x)=|x|+|x-1|$$ is

If $$f(x) = \sin^{-1} \left ( \frac{4^{x + \frac{1}{2}}}{1 + 2^{4x}} \right )$$, which of the following is not the derivative of f(x)?