At $x=2$ which utility function has the steepest slope?
A) $u(x)=\ln (x)$
B) $u(x)=\sqrt{x}$
C) $u(x)=.05 \mathrm{x}$
D) $u(x)=\frac{x^{4}}{.4}$

Question

Quizplus · Accepted Answer

To determine the steepest slope at $x=2$, we calculate the first derivative of each utility function with respect to $x$ and evaluate it at $x=2$. The function with the highest derivative value at $x=2$ has the steepest slope.A) $u'(x)=\frac{1}{x}$, so at $x=2$, $u'(2)=\frac{1}{2}$.B) $u'(x)=\frac{1}{2\sqrt{x}}$, so at $x=2$, $u'(2)=\frac{1}{2\sqrt{2}}$.C) $u'(x)=.05$, so at $x=2$, $u'(2)=.05$.D) $u'(x)=\frac{4x^{3}}{.4}$, so at $x=2$, $u'(2)=\frac{4(2)^{3}}{.4}=80$.Option D has the highest derivative value at $x=2$, indicating the steepest slope.

At x=2x=2x=2 Which Utility Function Has the Steepest Slope? A) u(x)=ln⁡(x)u(x)=\ln (x)u(x)=ln(x)

At $x=2$ Which Utility Function Has the Steepest Slope?
A) $u(x)=\ln (x)$