Question

Using Rolle's theorem, find points on the curve y = 16 − x², x ∈ [−1, 1], where tangent is parallel to x-axis.

Answer 1

The equation of the curve is

\[y = 16 - x^2\]  ...(1)

Let P\[\left( x_1 , y_1 \right)\] be a point on it where the tangent is parallel to the x-axis .

Then,

\[\left( \frac{dy}{dx} \right)_P = 0\]   ...(2)

Differentiating (1) with respect to x, we get

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

\[ \Rightarrow \left( \frac{dy}{dx} \right)_P = - 2 x_1 \]

\[ \Rightarrow - 2 x_1 = 0 \left( \text { from } \left( 2 \right) \right)\]

\[ \Rightarrow x_1 = 0\]

\[P\left( x_1 , y_1 \right)\]  lies on the curve\[y = 16 - x^2\] .

\[\therefore\] \[y_1 = 16 - {x_1}^2\]

When \[x_1 = 0\] ,

\[y_1 = 16\]

Hence,\[\left( 0, 16 \right)\] is the required point .