#### Question

Let f(x) = x^{3}+3x^{2 }\[-\] 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

\[\text { a minimum at x = 1}\]

\[\text { Given }: f\left( x \right) = x^3 + 3 x^2 - 9x + 2\]

\[ \Rightarrow f'\left( x \right) = 3 x^2 + 6x - 9\]

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

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

\[ \Rightarrow 3 x^2 + 6x - 9 = 0\]

\[ \Rightarrow x^2 + 2x - 3 = 0\]

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

\[ \Rightarrow x = - 3, 1\]

\[\text { Now,} \]

\[f''\left( x \right) = 6x + 6\]

\[ \Rightarrow f''\left( 1 \right) = 6 + 6 = 12 > 0\]

\[\text { So, x = 1 is a local minima } . \]

\[\text { Also }, \]

\[f''\left( - 3 \right) = - 18 + 6 = - 12 < 0\]

\[\text { So, x = - 3 is a local maxima } . \]