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Solution for Using Rolle'S Theorem, Find Points on the Curve Y = 16 − X2, X ∈ [−1, 1], Where Tangent is Parallel to X-axis. - CBSE (Commerce) Class 12 - Mathematics

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Question

Using Rolle's theorem, find points on the curve y = 16 − x2x ∈ [−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.
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