Application of Derivatives

The maximum distance from origin of a point on the curve $x = a\sin t - b\sin \left( {{{at} \over b}} \right)$ $y = a\cos t - b\cos \left( {{{at} \over b}} \right),$ both $a,b > 0$ is

The real number $x$ when added to its inverse gives the minimum sum at $x$ equal :

If the function $f\left( x \right) = 2{x^3} - 9a{x^2} + 12{a^2}x + 1,$ where $a>0,$ attains its maximum and minimum at $p$ and $q$ respectively such that ${p^2} = q$ , then $a$ equals

Area of the greatest rectangle that can be inscribed in the ellipse ${{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$

The function $f\left( x \right) = {x \over 2} + {2 \over x}$ has a local minimum at

A triangular park is enclosed on two sides by a fence and on the third side by a straight river bank. The two sides having fence are of same length $x$. The maximum area enclosed by the park is

If $p$ and $q$ are positive real numbers such that ${p^2} + {q^2} = 1$, then the maximum value of $(p+q)$ is

Suppose the cubic ${x^3} - px + q$ has three distinct real roots where $p>0$ and $q>0$. Then which one of the following holds?

The maximum volume (in cu.m) of the right circular cone having slant height 3 m is :

Given $P\left( x \right) = {x^4} + a{x^3} + b{x^2} + cx + d$ such that $x=0$ is the only real root of $P'\,\left( x \right) = 0.$ If $P\left( { - 1} \right) < P\left( 1 \right),$ then in the interval $\left[ { - 1,1} \right]:$