#### Question

The minimum value of f(x) = \[x4 - x2 - 2x + 6\] is _____________ .

6

4

8

none of these

#### Solution

`4`

\[\text { Given:} f\left( x \right) = x^4 - x^2 - 2x + 6\]

\[ \Rightarrow f'\left( x \right) = 4 x^3 - 2x - 2\]

\[ \Rightarrow f'\left( x \right) = \left( x - 1 \right)\left( 4 x^2 + 4x + 2 \right)\]

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

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

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

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

\[ \Rightarrow x = 1\]

\[\text { Now,} \]

\[f''\left( x \right) = 12 x^2 - 2\]

\[ \Rightarrow f''\left( 1 \right) = 12 - 2 = 10 > 0\]

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

\[\text { The local minimum value is given by }\]

\[f\left( 1 \right) = 1 - 1 - 2 + 6 = 4\]