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

Answer 1

None of these

 

\[\text { Let }\left( x_1 , y_1 \right)\text { be the required point.}\]

\[\text { Since, the point lies on the curve }, \]

\[\text { Hence }, y_1 = 12 x_1 - {x_1}^2 \]

\[\text { Now }, \]

\[y = 12x - x^2 \]

\[ \Rightarrow \frac{dy}{dx} = 12 - 2x\]

\[\text { Slope of the tangent } = \left( \frac{dy}{dx} \right)_\left( x_1 , y_1 \right) =12 - 2 x_1 \]

\[\text { Given }:\]

\[\text { Slope of the tangent }=0\]

\[12 - 2 x_1 = 0\]

\[ \Rightarrow x_1 = 6\]

\[\text { Now }, \]

\[ y_1 = 12 x_1 - {x_1}^2 = 72 - 36 = 36\]

\[ \therefore \left( x_1 , y_1 \right) = \left( 6, 36 \right)\]