Difficulty proving W(T) takes values on (0, K) and is strictly decreasing

[Jeff2017]

Hi All,

I have been spending more than an hour on this question and it is driving me nuts.

---

9. A theory of investment has used a function W defined for all T > 0 by:

. . . . .\(\displaystyle \displaystyle W(T)\, =\, \dfrac{K}{T}\, \int_0^T\, e^{-\varrho t}\, dt\)

...where K and \(\displaystyle \varrho\) are positive constants.

Evaluate the integral. Then prove that W(T) takes values on the interval (0, K) and is strictly decreasing. (Hint: Problem 6.11.11.)

---

So far I can only prove that the equation is strictly increasing instead of what the question is asking for (to prove it is decreasing).

 

[Deleted member 4993]

Hi All,

I have been spending more than an hour on this question and it is driving me nuts.

---

9. A theory of investment has used a function W defined for all T > 0 by:

. . . . .\(\displaystyle \displaystyle W(T)\, =\, \dfrac{K}{T}\, \int_0^T\, e^{-\varrho t}\, dt\)

...where K and \(\displaystyle \varrho\) are positive constants.

Evaluate the integral. Then prove that W(T) takes values on the interval (0, K) and is strictly decreasing. (Hint: Problem 6.11.11.)

---

So far I can only prove that the equation is strictly increasing instead of what the question is asking for (to prove it is decreasing).

Can you please show us the process by which you had proved W(T) is increasing?