# Thread: find number of positive roots of x = e^(ax for all real values of "a"

1. ## find number of positive roots of x = e^(ax for all real values of "a"

Questions and and b have already been answered. regarding c and d I'm kind of confused on how to get started with them. any suggestions?
Καταγραφή.jpg

2. ## find number of positive roots of x = e^(ax for all real values of "a"

Note: Parts (a) and (b) have already been solved. I'm struggling to move on to parts (c) and (d).

Given the following function:

. . . . .$f(x)\, =\, \begin{cases}x\, \cdot\, ln(x) & \mbox{ if }\, x\, >\, 0 \\ 0 & \mbox{ if }\, x\, =\, 0\end{cases}$

(a) Prove that $f(x)$ is continuous at $x\, =\, 0.$

(b) Study $f(x)$ as for the monotony, and find its domain.

(c) Find the number of the positive roots on the equation $x\, =\, e^{a/x}$ for all the real values of $a.$

(d) Prove $f'(x\, +\, 1)\, >\, f(x\, +\, 1)\, -\, f(x),\, \forall x\, >\, 0.$

My answers for parts (a) and (b):

(a) It is continuous at 0.

(b) $f(Df)\, =\, \left[-\dfrac{1}{e},\, \infty\right),$ at $\left(0,\, \dfrac{1}{e}\right]\, f(x)$ is a strictly decreasing function whereas at $\left[\dfrac{1}{e},\, \infty\right)\, f(x)$ is a strictly increasing function.

3. Originally Posted by Exodia
Note: Parts (a) and (b) have already been solved. I'm struggling to move on to parts (c) and (d).

Given the following function:

. . . . .$f(x)\, =\, \begin{cases}x\, \cdot\, ln(x) & \mbox{ if }\, x\, >\, 0 \\ 0 & \mbox{ if }\, x\, =\, 0\end{cases}$

(a) Prove that $f(x)$ is continuous at $x\, =\, 0.$

(b) Study $f(x)$ as for the monotony, and find its domain.

(c) Find the number of the positive roots on the equation $x\, =\, e^{a/x}$ for all the real values of $a.$

(d) Prove $f'(x\, +\, 1)\, >\, f(x\, +\, 1)\, -\, f(x),\, \forall x\, >\, 0.$

My answers for parts (a) and (b):

(a) It is continuous at 0.

(b) $f(Df)\, =\, \left[-\dfrac{1}{e},\, \infty\right),$ at $\left(0,\, \dfrac{1}{e}\right]\, f(x)$ is a strictly decreasing function whereas at $\left[\dfrac{1}{e},\, \infty\right)\, f(x)$ is a strictly increasing function.
You are asked to find roots of:

y = x - e^(a/x)

Assume some value of 'a' (= 0, 1, 2 etc) and plot the function. Look at the nature of the function.

Tell us what you found....