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Question Number 214335 by depressiveshrek last updated on 05/Dec/24

$$\mathrm{For}\mathrm{what}\mathrm{values}\mathrm{of}k\mathrm{does}\mathrm{the}\mathrm{equation}$$$$e^{kx}=3\sqrt{x}\mathrm{have}\mathrm{only}\mathrm{one}\mathrm{solution}\mathrm{in}\mathbb{R}?$$

Answered by mr W last updated on 06/Dec/24

$$y=e^{kx}andy=3\sqrt{x}shouldhaveonly$$$$onetouchingpoint,sayat(p,q).$$$$then{e}^{kx}=3\sqrt{x}hasonlyonereal$$$$solutionx=p.$$$$$$$$y=e^{kx}$$$$\frac{dy}{dx}={ke}^{kx}$$$$y=3\sqrt{x}$$$$\frac{dy}{dx}=\frac{3}{2\sqrt{x}}$$$$q=e^{kp}=3\sqrt{p}...\left(i\right)$$$${ke}^{kp}=\frac{3}{2\sqrt{p}}...\left(ii\right)$$$$k\times 3\sqrt{p}=\frac{3}{2\sqrt{p}}\Rightarrow kp=\frac{1}{2}$$$$e^{\frac{1}{2}}=3\sqrt{p}\Rightarrow p=\frac{e}{9}$$$$\Rightarrow k=\frac{1}{2p}=\frac{9}{2e}✓$$

Commented by mr W last updated on 06/Dec/24

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