MAT334-2018F > Quiz-5

Q5 TUT 0102

(1/1)

Victor Ivrii:
Give the order of each of the zeros of the given function:
$$e^{2z}-3e^z-4.$$

Xiting Kuang:

Yuechen Huang:

f(z) = e^z - 3e^z - 4 = 0

Let $w=e^z$, then

w^2 - 3w - 4 = 0 \\
(w-4)(w+1) = 0 \Rightarrow w = 4 \space or \space w = 1 \\
e^z = 4 \space or \space e^z = -1 \\
z = \log4 \space or \space z = \log(-1) \\

When $e^z = 4$, the order is 1

f'(z) =2e^{2z} - 3e^{z} = 2 \times 4^2 - 3 \times 4 \neq 0

When $e^z = -1$, the order is 1

f'(z) =2e^{2z} - 3e^{z} = 2 \times (-1)^2 - 3 \times (-1) \neq 0

Victor Ivrii:
Yuechen
you need to write what is $\log(4)$ and $\log(-1)$