Toronto Math Forum
MAT3342018F => MAT334Tests => Quiz5 => Topic started by: Victor Ivrii on November 02, 2018, 03:29:27 PM

Give the order of each of the zeros of the given function:
$$
e^{2z}3e^z4.
$$

Please see the attachment below.

\begin{equation}
f(z) = e^z  3e^z  4 = 0
\end{equation}
Let $w=e^z$, then
\begin{equation}
w^2  3w  4 = 0 \\
(w4)(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) \\
\end{equation}
When $e^z = 4$, the order is 1
\begin{equation}
f'(z) =2e^{2z}  3e^{z} = 2 \times 4^2  3 \times 4 \neq 0
\end{equation}
When $e^z = 1$, the order is 1
\begin{equation}
f'(z) =2e^{2z}  3e^{z} = 2 \times (1)^2  3 \times (1) \neq 0
\end{equation}

Yuechen
you need to write what is $\log(4)$ and $\log(1)$