Question

Ordinate of the point on the curve y=x^x where tangent drawn to the curve is parallel to x axis is a)e^(1/e) b) 1/e c) e^(-1/e) d) 1/(e^e)

Answer 1

$$\begin{gathered} \mathrm{Dear}\mathrm{Student}, \\ \mathrm{y}={\mathrm{x}}^{\mathrm{x}} \\ \mathrm{Taking}\ln , \\ \mathrm{lny}=\mathrm{xlnx} \\ \mathrm{differentiating}\mathrm{both}\mathrm{side}, \\ \frac{1}{\mathrm{y}}\frac{\mathrm{dy}}{\mathrm{dx}}=1+\mathrm{lnx} \\ \frac{\mathrm{dy}}{\mathrm{dx}}=\left(1+\mathrm{lnx}\right){\mathrm{x}}^{\mathrm{x}} \\ \mathrm{Now}\mathrm{if}\mathrm{slop}\mathrm{is}\mathrm{parallel}\mathrm{to}\mathrm{x}-\mathrm{axis}\mathrm{then} \\ \frac{\mathrm{dy}}{\mathrm{dx}}=0 \\ \left(1+\mathrm{lnx}\right){\mathrm{x}}^{\mathrm{x}}=0 \\ \left(1+\mathrm{lnx}\right)=0 \\ \mathrm{lnx}=-1 \\ \mathrm{x}={\mathrm{e}}^{-1}=\frac{1}{\mathrm{e}} \\ \mathrm{Regards} \end{gathered}$$