Questions: Consider the curve defined by (x^2=e^x-y) for (x>0). At what value of (x) does the curve have a horizontal tangent?
(A) 1
(B) (sqrt2)
C 2
D There is no such value of (x).
Transcript text: Consider the curve defined by $x^{2}=e^{x-y}$ for $x>0$. At what value of $x$ does the curve have a horizontal tangent?
(A) 1
(B) $\sqrt{2}$
C 2
D There is no such value of $x$.
Solution
To find the value of \( x \) where the curve \( x^2 = e^{x-y} \) has a horizontal tangent, we need to determine where the derivative of \( y \) with respect to \( x \) is zero. This involves implicit differentiation of the given equation and solving for \( \frac{dy}{dx} = 0 \).
Step 1: Implicit Differentiation
Given the curve \( x^2 = e^{x-y} \), we differentiate both sides with respect to \( x \) implicitly:
\[
\frac{d}{dx}(x^2) = \frac{d}{dx}(e^{x-y})
\]
\[
2x = e^{x-y} \left(1 - \frac{dy}{dx}\right)
\]