Maxima and MinimaHard

Question

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

Options

A.-81
B.-40
C.-41
D.-80

Solution

$f(x) = x^{2025} - x^{2000}$

$f'(x) = 0 \Rightarrow x = \left( \frac{2000}{2025} \right)^{1/25} = \alpha($ say $)$

$$\therefore f(0) = 0,f(1) = 0,f(\alpha) = \left( \frac{80}{81} \right)^{80} \cdot \frac{- 1}{81} = 80^{80} \cdot ( - 81)^{- 81}$$

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