Let $f:[1,3] \rightarrow R$ be a continuous function that is differentiable in $(1,3)$ and $f^{\prime}(x)=|f(x)|^{2}+4$ for all $x \in(1,3).$ Then,

If $L.M.V.T.$ is true for $f(x) = x(x-1)(x-2)$ on the interval $x \in [0, 1/2]$,then find the value of $C$.

The function $f(x) = x^3 - 6x^2 + ax + b$ satisfies the conditions of Rolle's theorem in $[1, 3]$. Then the values of $a$ and $b$ are respectively

Let $f: R \rightarrow R$ be a thrice differentiable function such that $f(0)=0, f(1)=1, f(2)=-1, f(3)=2$ and $f(4)=-2$. Then,the minimum number of zeros of $(3 f^{\prime} f^{\prime \prime} + f f^{\prime \prime \prime})(x)$ is....................

Let $f: R \rightarrow R$ be a twice continuously differentiable function. Let $f(0)=f(1)=f^{\prime}(0)=0$. Then,

Consider $f(x) = \int\limits_0^x {\left( {t + \frac{1}{t}} \right)\,dt}$ and $g(x) = f'(x)$ for $x \in \left[ {\frac{1}{2}, 3} \right]$. If $P$ is a point on the curve $y = g(x)$ such that the tangent to this curve at $P$ is parallel to a chord joining the points $\left( {\frac{1}{2}, g\left( {\frac{1}{2}} \right)} \right)$ and $(3, g(3))$ of the curve,then the coordinates of the point $P$ are: