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

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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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