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Question
f(x) = x3 \[-\] 3x.
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Solution
\[\text { Given: } \hspace{0.167em} f\left( x \right) = \left( x - 5 \right)^4 \]
\[ \Rightarrow f'\left( x \right) = 4 \left( x - 5 \right)^3 \]
\[\text { For a local maximum or a local minimum, we must have }\]
\[f'\left( x \right) = 0\]
\[ \Rightarrow 4 \left(x - 5 \right)^3 = 0\]
\[ \Rightarrow x = 5\]
Since f '(x) changes from negative to positive as x increases through 1, x = 1 is the point of local minima.
The local minimum value of f (x) at x = 1 is given by \[\left( 1 \right)^3 - 3\left( 1 \right) = - 2\]
Since f '(x) changes from positive to negative when x increases through -1, x = -1 is the point of local maxima.
The local maximum value of f (x) at x = -1 is given by \[\left( - 1 \right)^3 - 3\left( - 1 \right) = 2\]
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