If the minimum value of $f(x) = \frac{5x^2}{2} + \frac{\alpha}{x^5}$ for $x > 0$ is $14$,then the value of $\alpha$ is equal to:

The maximum value $M$ of $f(x) = 3^x + 5^x - 9^x + 15^x - 25^x$,as $x$ varies over all real numbers,satisfies:

The values of $x$ at the stationary points of $f(x)=x^3+3x^2-2$ are

$A$ tank with a rectangular base and rectangular sides, open at the top is made. The depth of the tank is $4 \text{ m}$ and its volume is $36 \text{ m}^3$. The cost of the base material is $\text{Rs. } 100 \text{ per m}^2$ and the cost of the side material is $\text{Rs. } 50 \text{ per m}^2$. Find the minimum cost of the tank.

The function $f(x) = \frac{x^2 - 2}{\sqrt{1 + x^2}}$:

If the function $f(x)=2x^3-9ax^2+12a^2x+1, a>0$ has a local maximum at $x=\alpha$ and a local minimum at $x=\alpha^2$,then $\alpha$ and $\alpha^2$ are the roots of the equation: