PUC Karnataka Science Class 12Department of Pre-University Education, Karnataka
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Solution for At What Point on the Following Curve, is the Tangent Parallel to X-axis Y = E 1 − X 2 on [−1, 1] ? - PUC Karnataka Science Class 12 - Mathematics

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Question

At what point  on the following curve, is the tangent parallel to x-axis y = \[e^{1 - x^2}\] on [−1, 1] ?

Solution

\[f\left( x \right) = e^{1 - x^2}\]

Since 

\[f\left( x \right)\] is an exponential function, which is continuous and derivable on its domain,

\[f\left( x \right)\] is continuous on \[\left[ - 1, 1 \right]\] and differentiable on \[\left( - 1, 1 \right)\].
Also, 
\[f\left( 1 \right) = f\left( - 1 \right) = 1\]
Thus, all the conditions of Rolle's theorem are satisfied.
Consequently, there exists at least one point c
\[\in \left( - 1, 1 \right)\] for which  \[f'\left( c \right) = 0\] .
But 
\[f'\left( c \right) = 0 \Rightarrow - 2c e^{1 - c^2} = 0 \Rightarrow c = 0 \left( \because e^{1 - c^2} \neq 0 \right)\]
\[\therefore f\left( c \right) = f\left( 0 \right) = e\] 
By the geometrical interpretation of Rolle's theorem, \[\left( 0, e \right)\] is the point on \[y = e^{1 - x^2}\]  where the tangent is parallel to the x-axis .
  Is there an error in this question or solution?
Solution At What Point on the Following Curve, is the Tangent Parallel to X-axis Y = E 1 − X 2 on [−1, 1] ? Concept: Maximum and Minimum Values of a Function in a Closed Interval.
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