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प्रश्न
The least and greatest values of f(x) = x3\[-\] 6x2+9x in [0,6], are ___________ .
पर्याय
3, 4
0, 6
0, 3
3, 6
0, 54
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उत्तर
\[\text { Given: } f\left( x \right) = x^3 - 6 x^2 + 9x\]
\[ \Rightarrow f'\left( x \right) = 3 x^2 - 12x + 9\]
\[\text { For a local maxima or a local minima, we must have }\]
\[f'\left( x \right) = 0\]
\[ \Rightarrow 3 x^2 - 12x + 9 = 0\]
\[ \Rightarrow x^2 - 4x + 3 = 0\]
\[ \Rightarrow \left( x - 1 \right)\left( x - 3 \right) = 0\]
\[ \Rightarrow x = 1, 3\]
\[\text { Now,} \]
\[f\left( 0 \right) = 0^3 - 6 \left( 0 \right)^2 + 9\left( 0 \right) = 0\]
\[f\left( 1 \right) = 1^3 - 6 \left( 1 \right)^2 + 9\left( 1 \right) = 1 - 6 + 9 = 4\]
\[f\left( 3 \right) = 3^3 - 6 \left( 3 \right)^2 + 9\left( 3 \right) = 27 - 54 + 27 = 0\]
\[f\left( 6 \right) = 6^3 - 6 \left( 6 \right)^2 + 9\left( 6 \right) = 216 - 216 + 54 = 54\]
The least and greatest values of f(x) = x3- 6x2+9x in [0, 6] are 0 and 54, respectively.
Notes
The question in the book has some error. So, none of the options are matching with the solution. The solution here is according to the question given in the book.
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