The maximum value of $x^4 y^4$ given the constraint $a^2 x^4 + b^2 y^4 = c^6$ is

Let $AD$ and $BC$ be two vertical poles at $A$ and $B$ respectively on a horizontal ground. If $AD = 8 \ m$,$BC = 11 \ m$ and $AB = 10 \ m$; then the distance (in meters) of a point $M$ on $AB$ from the point $A$ such that $MD^{2} + MC^{2}$ is minimum is

Let $f: R \rightarrow R$ be a polynomial function of degree four having extreme values at $x=4$ and $x=5$. If $\lim _{x \rightarrow 0} \frac{f(x)}{x^2}=5$,then $f(2)$ is equal to:

The range of $a \in R$ for which the function $f(x)=(4 a-3)\left(x+\log _{e} 5\right)+2(a-7) \cot \left(\frac{x}{2}\right) \sin ^{2}\left(\frac{x}{2}\right)$ where $x \neq 2 n \pi, n \in N$,has critical points,is

The sum of the lengths of the hypotenuse and one side of a right-angled triangle is constant. The area of the triangle will be maximum if the angle between them is:

If $a, b > 0$,then the minimum value of $y = \frac{b^2}{a-x} + \frac{a^2}{x}$ for $0 < x < a$ is