A value of $c$ for which the conclusion of mean value the theorem holds for the function $f(x) = log{_e}x$ on the interval $[1, 3]$ is
$log_e\ 3$
$log_3\ e$
$2\ log_3\ e$
$\frac{1}{2}{\log _e}\,3$
If $f:[-5,5] \rightarrow \mathrm{R}$ is a differentiable function and if $f^{\prime}(x)$ does not vanish anywhere, then prove that $f(-5) \neq f(5).$
The function $f(x) = x(x + 3){e^{ - (1/2)x}}$ satisfies all the conditions of Rolle's theorem in $ [-3, 0]$. The value of $c$ is
If $f:R \to R$ and $f(x)$ is a polynomial function of degree ten with $f(x)=0$ has all real and distinct roots. Then the equation ${\left( {f'\left( x \right)} \right)^2} - f\left( x \right)f''\left( x \right) = 0$ has
Consider $f (x) = | 1 - x | \,;\,1 \le x \le 2 $ and $g (x) = f (x) + b sin\,\frac{\pi }{2}\,x$, $1 \le x \le 2$ then which of the following is correct ?
Let $f (x)$ and $g (x)$ are two function which are defined and differentiable for all $x \ge x_0$. If $f (x_0) = g (x_0)$ and $f ' (x) > g ' (x)$ for all $x > x_0$ then