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Question
The point at the curve y = 12x − x2 where the slope of the tangent is zero will be _____________ .
Options
(0, 0)
(2, 16)
(3, 9)
none of these
Solution
None of these
\[\text { Let }\left( x_1 , y_1 \right)\text { be the required point.}\]
\[\text { Since, the point lies on the curve }, \]
\[\text { Hence }, y_1 = 12 x_1 - {x_1}^2 \]
\[\text { Now }, \]
\[y = 12x - x^2 \]
\[ \Rightarrow \frac{dy}{dx} = 12 - 2x\]
\[\text { Slope of the tangent } = \left( \frac{dy}{dx} \right)_\left( x_1 , y_1 \right) =12 - 2 x_1 \]
\[\text { Given }:\]
\[\text { Slope of the tangent }=0\]
\[12 - 2 x_1 = 0\]
\[ \Rightarrow x_1 = 6\]
\[\text { Now }, \]
\[ y_1 = 12 x_1 - {x_1}^2 = 72 - 36 = 36\]
\[ \therefore \left( x_1 , y_1 \right) = \left( 6, 36 \right)\]
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