Answer
(a) The maximum value is approximately $2.85$
The minimum value is approximately $1.90$
(b) The maximum value is $2.85$
The minimum value is $1.89$
Work Step by Step
(a) $f(x) = e^x+e^{-2x}$
We can use the zoom function on a graphing calculator to estimate the maximum and minimum values of the function in the interval $[0,1]$
The maximum value is approximately $2.85$
The minimum value is approximately $1.90$
(b) $f(x) = e^x+e^{-2x}$
$f'(x) = e^x-2e^{-2x} = 0$
$e^x = 2e^{-2x}$
$e^{3x} = 2$
$3x = ln(2)$
$x = \frac{ln(2)}{3}$
$x = 0.231$
When $x = 0.231$:
$f(0.231) = e^{0.231}+e^{-(2)(0.231)}= 1.89$
The minimum value is $1.89$
When $x = 1$:
$f(1) = e^{1}+e^{-(2)(1)}= 2.85$
The maximum value is $2.85$
