For-what-values-of-k-does-the-equation-e-kx-3-x-have-only-one-solution-in-R-

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

For what values of k does the equation

e^{k x} = 3 \sqrt{x} have only one solution in R ?

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

y = e^{k x} a n d y = 3 \sqrt{x} s h o u l d h a v e o n l y

o n e t o u c h i n g p o i n t, s a y a t (p, q) .

t h e n e^{k x} = 3 \sqrt{x} h a s o n l y o n e r e a l

s o l u t i o n x = p .

y = e^{k x}

\frac{d y}{d x} = {k e}^{k x}

y = 3 \sqrt{x}

\frac{d y}{d x} = \frac{3}{2 \sqrt{x}}

q = e^{k p} = 3 \sqrt{p} \dots (i)

{k e}^{k p} = \frac{3}{2 \sqrt{p}} \dots (i i)

k \times 3 \sqrt{p} = \frac{3}{2 \sqrt{p}} \Rightarrow k p = \frac{1}{2}

e^{\frac{1}{2}} = 3 \sqrt{p} \Rightarrow p = \frac{e}{9}

\Rightarrow k = \frac{1}{2 p} = \frac{9}{2 e} ✓

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

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