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The Least and Greatest Values of F(X) = X3 − 6x2+9x in [0,6], Are (A) 3,4 (B) 0,6 (C) 0,3 (D) 3,6 - CBSE (Science) Class 12 - Mathematics

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Question

The least and greatest values of f(x) = x3\[-\] 6x2+9x in [0,6], are ___________ .

  • 3, 4

  • 0, 6

  • 0, 3

  • 3, 6

  • 0, 54

Solution

\[\text { Given: } f\left( x \right) = x^3 - 6 x^2 + 9x\]

\[ \Rightarrow f'\left( x \right) = 3 x^2 - 12x + 9\]

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

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

\[ \Rightarrow 3 x^2 - 12x + 9 = 0\]

\[ \Rightarrow x^2 - 4x + 3 = 0\]

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

\[ \Rightarrow x = 1, 3\]

\[\text { Now,} \]

\[f\left( 0 \right) = 0^3 - 6 \left( 0 \right)^2 + 9\left( 0 \right) = 0\]

\[f\left( 1 \right) = 1^3 - 6 \left( 1 \right)^2 + 9\left( 1 \right) = 1 - 6 + 9 = 4\]

\[f\left( 3 \right) = 3^3 - 6 \left( 3 \right)^2 + 9\left( 3 \right) = 27 - 54 + 27 = 0\]

\[f\left( 6 \right) = 6^3 - 6 \left( 6 \right)^2 + 9\left( 6 \right) = 216 - 216 + 54 = 54\]

The least and greatest values of f(x) = x3- 6x2+9x in [0, 6] are 0 and 54, respectively.

Notes

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Solution The Least and Greatest Values of F(X) = X3 − 6x2+9x in [0,6], Are (A) 3,4 (B) 0,6 (C) 0,3 (D) 3,6 Concept: Graph of Maxima and Minima.
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