Maxima and Minima

The maximum value of $(\cos\alpha_1)(\cos\alpha_2)\cdots(\cos\alpha_n)$ where $0\le \alpha_1,\alpha_2,\ldots,\alpha_n\le\pi$ and $(\cot\alpha_1)(\cot\alpha_2)\cdots(\cot\alpha_n)=1$ is

The function $f(x)=2\sin x+\sin 2x,\ x\in[0,2\pi]$ has absolute maximum and minimum at

A box open at the top is made by cutting squares from the four corners of a $6 \times 6$ m sheet. The height of the box for maximum volume is:

The minimum value of the function $y=2x^{3}+36x-20$ is

Two non-negative numbers whose sum is 9 and the product of the one number and square of the other number is maximum, are

A condition that $x^{3} + ax^{2} + bx + c$ may have no extremum is

A function $f : (0,\pi) \to R$ defined by $f(x) = 2 sin x + cos 2x$ has

The critical point and nature for the function f(x, y) = x² –2x + 2y² + 4y – 2 is

The maximum value of 4 sin² x + 3 cos² x + sin(x/2) + cos(x/2) is

In an acute-angled ΔABC the least value of secA+secB+secC is

If A > 0, B > 0 and A + B = $\frac{\pi}{6}$ , then the minimum value of $ \tan A + \tan B$

The sum of infinite terms of a decreasing GP is equal to the greatest value of the function $f(x)=x^3+3x-9$ in the interval [-2,3] and the difference between the first two terms is $f'(0)$. Then the common ratio of GP is

Find the interval(s) on which the graph y=2x³e^x is increasing

The function $f(x)=\frac{x}{1+x\tan x}$ , $0\leq x\leq\frac{\pi}{2}$ is maximum when