Question

The equation of the tangent to the curve \(y = \sqrt{9-3x^2}\) at the point where the ordinate and abscissa equal is...

• A. \(x-3y+3 = 0\)
• B. \(3x-y-3 = 0\)
• C. \(x+3y-6 = 0\)
• D. \(3x+y-6 = 0\)

Hint:

Take logarithms and find where the derivative of -x ln x vanishes.

Answer 1

The Correct Option is D

Step 1: Direct differentiation:
$y = x^{-x}$, $\frac{dy}{dx} = x^{-x}(-\ln x - 1)$. This changes from positive to negative at $x = \frac1e$, so it is a maximum.

Step 2: Evaluate:
$y\left(\frac1e\right) = \left(\frac1{1/e}\right)^{1/e} = e^{1/e}$ (C).

Final Answer:
$e^{1/e}$. \[ \boxed{e^{1/e}} \]