Application of Derivatives

The minimum value of $(u-v)^2+\left(\sqrt{2-u^2}-\frac{9}{v}\right)^2$, where $0 < u < \sqrt{2}$ and $v > 0$

The function, $f(x)=(3 x-7) x^{2 / 3}, x \in R$ is increasing for all $x$ lying in

Let $f(x)$ be a polynomial of degree 6 divisible by $x^3$ and having a point of extremum at $x=2$. If $f^{\prime}(x)$ is divisible by $1+x^2$, then find the value of $\frac{3 f(2)}{f(1)}$

The length of three sides of a trapezium are equal, each being 10 cms . Then, the maximum area $\left(\mathrm{cm}^2\right)$ of the trapezium is

If tangent to the curve $f(x)=x^3-\alpha x^2-x+\beta$ at point $(1,3)$ on the curve, cut equals non zero intercepts on co-ordinate axes, then

Difference between the maximum and minimum values of $f(x)=-\sin ^3 x+3 \sin ^2 x+5$ in $x \in\left[0, \frac{\pi}{2}\right]$ is

The maximum slope of the curve \(y=\frac{1}{2} x^4-5 x^3+18 x^2-19 x+7\) occurs at the point

\(f\) and \(g\) are differentiable function in \((0,1)\) satisfying \(f(0)=2=g(\mathrm{l}), g(0)=0\) and \(f(l)=6\), then for some \(c \in] 0,1[\)

The maximum slope of the curve \(y=\frac{1}{2} x^4-5 x^3+18 x^2-19 x\) occurs at the point

The slope of normal to the curve \(y=x^3+2 x+6\) which is parallel to line \(x+14 y+4=0\) is

The interval in which the function \(f(x)=\sin x-\cos x, 0 \leq x \leq 2 \pi\) is strictly decreasing, is