Let $f(x) = x^{2025} - x^{2000}$, $x \in [0, 1]$ and the minimum value of the function $f(x)$ in the interval $[0, 1]$ be $(80)^{80}(n)^{-81}$. Then $n$ is equal to

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 difference between the local extreme values of the function $f(x) = 2x^3 - 15x^2 + 36x + 40$ is ......

$A$ tank with a rectangular base and rectangular sides,open at the top,is to be constructed such that its depth is $2 \ m$ and volume is $8 \ m^{3}$. If the cost of building the tank is Rs $70$ per square metre for the base and Rs $45$ per square metre for the sides,what is the minimum cost of the tank?

$A$ wire of length $8 \text{ units}$ is cut into two parts which are bent respectively in the form of a square and a circle. The least value of the sum of the areas so formed is

The sum of the terms of an infinitely decreasing geometric progression is equal to the greatest value of the function $f(x) = x^3 + 3x - 9$ on the interval $[-2, 3]$. If the difference between the first and the second term of the progression is equal to $f'(0)$,then the common ratio of the $G.P.$ is