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Solution for F(X) = X3 (2x − 1)3. - CBSE (Science) Class 12 - Mathematics

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Question

f(x) = x3 (2x \[-\] 1)3.

Solution

\[\text { Given:} f\left( x \right) = x^3 \left( 2x - 1 \right)^3 \]

\[ \Rightarrow f'\left( x \right) = 3 x^2 \left( 2x - 1 \right)^3 + 6 x^3 \left( 2x - 1 \right)^2 \]

\[\text { For the local maxima or minima, we must have }\]

\[ f'\left( x \right) = 0\]

\[ \Rightarrow 3 x^2 \left( 2x - 1 \right)^3 + 6 x^3 \left( 2x - 1 \right)^2 = 0\]

\[ \Rightarrow 3 x^2 \left( 2x - 1 \right)^2 \left( 2x - 1 + 2x \right) = 0\]

\[ \Rightarrow x^2 \left( 2x - 1 \right)^2 \left( 4x - 1 \right) = 0\]

\[ \Rightarrow x = 0, \frac{1}{2} \text { and } \frac{1}{4}\]

Sincef '(x) changes from negative to positive when x increases through \[\frac{1}{4}\] x = \[\frac{1}{4}\] is a point of local minima.
The local minimum value of  f (x)  at x = \[\frac{1}{4}\] is given by

\[\left( \frac{1}{4} \right)^3 \left( \frac{1}{2} - 1 \right)^3 = \frac{- 1}{512}\]
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Solution F(X) = X3 (2x − 1)3. Concept: Graph of Maxima and Minima.
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