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

ConceptMaximum and Minimum Values of a Function in a Closed Interval

#### 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?

#### APPEARS IN

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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