The abscissa of the point on the curve y=alef | vedclass

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(A) Given the curve equation: $y = a\left(e^{\frac{x}{a}} + e^{-\frac{x}{a}}\right)$.To find the slope of the tangent,we differentiate $y$ with respec

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$0$

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(A) Given the curve equation: $y = a\left(e^{\frac{x}{a}} + e^{-\frac{x}{a}}\right)$.To find the slope of the tangent,we differentiate $y$ with respect to $x$:$\frac{dy}{dx} = a \left( e^{\frac{x}{a}} \cdot \frac{1}{a} + e^{-\frac{x}{a}} \cdot (-\frac{1}{a}) \right) = e^{\frac{x}{a}} - e^{-\frac{x}{a}}$.Since the tangent is parallel to the $X$-axis,its slope must be zero:$\frac{dy}{dx} = 0 \implies e^{\frac{x}{a}} - e^{-\frac{x}{a}} = 0$.$e^{\frac{x}{a}} = e^{-\frac{x}{a}}$.Taking the natural logarithm on both sides:$\frac{x}{a} = -\frac{x}{a} \implies \frac{2x}{a} = 0 \implies x = 0$.Thus,the abscissa of the point is $0$.

The abscissa of the point on the curve $y=a\left(e^{\frac{x}{a}}+e^{-\frac{x}{a}}\right)$ where the tangent is parallel to the $X$-axis is

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