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Question
Using Rolle's theorem, find points on the curve y = 16 − x2, x ∈ [−1, 1], where tangent is parallel to x-axis.
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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 .
\[\frac{dy}{dx} = - 2x\]
\[ \Rightarrow \left( \frac{dy}{dx} \right)_P = - 2 x_1 \]
\[ \Rightarrow - 2 x_1 = 0 \left( \text { from } \left( 2 \right) \right)\]
\[ \Rightarrow x_1 = 0\]
Hence,\[\left( 0, 16 \right)\] is the required point .
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