PUC Karnataka Science Class 12Department of Pre-University Education, Karnataka
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# The Minimum Value of F(X) = X 4 − X 2 − 2 X + 6 is (A) 6 (B) 4 (C) 8 (D) None of These - PUC Karnataka Science Class 12 - Mathematics

#### Question

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

• 6

• 4

• 8

• none of these

#### Solution

4

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

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

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

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

$f'\left( x \right) = 0$

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

$\Rightarrow \left( x - 1 \right) = 0$

$\Rightarrow x = 1$

$\text { Now,}$

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

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

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

$\text { The local minimum value is given by }$

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

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Solution The Minimum Value of F(X) = X 4 − X 2 − 2 X + 6 is (A) 6 (B) 4 (C) 8 (D) None of These Concept: Graph of Maxima and Minima.
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