If the function $f$ is given by $f(x)=x^3-3(a-2)x^2+3ax+7$,for some $a \in R$,is increasing in $(0,1]$ and decreasing in $[1,5)$,then a root of the equation $\frac{f(x)-14}{(x-1)^2}=0$ $(x \neq 1)$ is

The maximum value of $f(x) = x^2 \log x$ on the interval $[1, e]$ is:

The shortest distance between the point $\left( \frac{3}{2}, 0 \right)$ and the curve $y = \sqrt{x}, (x > 0)$ is:

Let $AP$ and $BQ$ be two vertical poles at points $A$ and $B$,respectively. If $AP=16 \, m, BQ=22 \, m$ and $AB=20 \, m,$ then find the distance of a point $R$ on $AB$ from the point $A$ such that $RP^2 + RQ^2$ is minimum. (in $, m$)

If the derivative of a function $f$ is given by $f'(x) = (x - a)^{2m} (x - b)^{2n + 1}$,where $m$ and $n$ are positive integers and $a > b$,then which of the following is true?

Find the local maximum and local minimum values for the function $h(x) = \sin x + \cos x$ in the interval $0 < x < \frac{\pi}{2}$.