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
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
Last edited by stapel; 03-08-2018 at 01:03 PM. Reason: Reinserting original post text.
Note: Parts (a) and (b) have already been solved. I'm struggling to move on to parts (c) and (d).
Given the following function:
. . . . .[tex]f(x)\, =\, \begin{cases}x\, \cdot\, ln(x) & \mbox{ if }\, x\, >\, 0 \\ 0 & \mbox{ if }\, x\, =\, 0\end{cases}[/tex]
(a) Prove that [tex]f(x)[/tex] is continuous at [tex]x\, =\, 0.[/tex]
(b) Study [tex]f(x)[/tex] as for the monotony, and find its domain.
(c) Find the number of the positive roots on the equation [tex]x\, =\, e^{a/x}[/tex] for all the real values of [tex]a.[/tex]
(d) Prove [tex]f'(x\, +\, 1)\, >\, f(x\, +\, 1)\, -\, f(x),\, \forall x\, >\, 0.[/tex]
My answers for parts (a) and (b):
(a) It is continuous at 0.
(b) [tex]f(Df)\, =\, \left[-\dfrac{1}{e},\, \infty\right),[/tex] at [tex]\left(0,\, \dfrac{1}{e}\right]\, f(x)[/tex] is a strictly decreasing function whereas at [tex]\left[\dfrac{1}{e},\, \infty\right)\, f(x)[/tex] is a strictly increasing function.
Last edited by stapel; 03-08-2018 at 01:04 PM. Reason: Typing out the text in the graphic.
Last edited by stapel; 03-08-2018 at 01:05 PM. Reason: Copying typed-out graphical content into reply.
“... mathematics is only the art of saying the same thing in different words” - B. Russell
Bookmarks