Given that curve $C$ is defined by ${{y}^{x}}=x$ for $x0$, Verify that $C$ has a stationary point a

Question

Given that curve $C$ is defined by ${{y}^{x}}=x$ for $x>0$, Verify that $C$ has a stationary point at $x=\mathrm{e}$. [5] Show that $x\frac{{{\mathrm{d}}^{2}}y}{\mathrm{d}{{x}^{2}}}+\frac{\mathrm{d}y}{\mathrm{d}x}\left( 2+\ln y-\frac{1}{x} \right)=-\frac{y}{{{x}^{2}}}$ and deduce the nature of the stationary point found in (i). [5]

Accepted Answer

(ii) $\frac{{{\mathrm{d}}^{2}}y}{\mathrm{d}{{x}^{2}}}=-\frac{{{\mathrm{e}}^{\frac{1}{\mathrm{e}}}}}{{{\mathrm{e}}^{3}}}<0$. The stationary point is a maximum point.

2022 TJC J1 MYE Q7

2022 TJC J1 MYE Q7

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