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

Exodia said:

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

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Given the following function:

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

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

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

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

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

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

(a) It is continuous at 0.

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

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