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Question
Let f(x) = x3+3x2 \[-\] 9x+2. Then, f(x) has _________________ .
Options
a maximum at x = 1
a minimum at x = 1
neither a maximum nor a minimum at x = - 3
none of these
Solution
\[\text { a minimum at x = 1}\]
\[\text { Given }: f\left( x \right) = x^3 + 3 x^2 - 9x + 2\]
\[ \Rightarrow f'\left( x \right) = 3 x^2 + 6x - 9\]
\[\text { For a local maxima or a local minima, we must have } \]
\[f'\left( x \right) = 0\]
\[ \Rightarrow 3 x^2 + 6x - 9 = 0\]
\[ \Rightarrow x^2 + 2x - 3 = 0\]
\[ \Rightarrow \left( x + 3 \right)\left( x - 1 \right) = 0\]
\[ \Rightarrow x = - 3, 1\]
\[\text { Now,} \]
\[f''\left( x \right) = 6x + 6\]
\[ \Rightarrow f''\left( 1 \right) = 6 + 6 = 12 > 0\]
\[\text { So, x = 1 is a local minima } . \]
\[\text { Also }, \]
\[f''\left( - 3 \right) = - 18 + 6 = - 12 < 0\]
\[\text { So, x = - 3 is a local maxima } . \]
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