First slide

Mean value theorems

Question

Let the x co-ordinate of a point P on the function $f (x) = 2 x {(x - 3)}^{n} o n [0, 3]$ for which Rolle’s theorem is valid is $\frac{3}{4}$ . Then find 100n is,

Moderate

A

30
B

303
C

300
D

100

Solution

$\begin{aligned} f (x) = 2 x (x - 3)^{n} \\ f^{'} (x) = 2 x n (x - 3)^{n - 1} + (x - 3) \times 2 \\ f^{'} (c) = 0 \Rightarrow 2 (c - 3)^{n - 1} [(c - 3) + n c] = 0 \\ Given \begin{array}{l} c = \frac{3}{4}, (n + 1) \frac{3}{4} = 3 \\ \underline{n} = 3 \end{array} \end{aligned}$

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