#### Question

Using Rolle's theorem, find points on the curve *y* = 16 − *x*^{2}, *x* ∈ [−1, 1], where tangent is parallel to *x*-axis.

#### Solution

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 .

Is there an error in this question or solution?

Solution Using Rolle'S Theorem, Find Points on the Curve Y = 16 − X2, X ∈ [−1, 1], Where Tangent is Parallel to X-axis. Concept: Maximum and Minimum Values of a Function in a Closed Interval.