Question

1. submit answer practice similar attempt 1: 10 attempts remaining. find the absolute maximum and minimum y - values of f(x)=x - 5ln(x) on the interval 1/5,10, as well as the x - value(s) where they occur. if there are multiple answers, separate your answers by a comma. if they do not exist, enter dne. round your answers to 4 decimal places if necessary. (a) absolute maximum of y = occurring at x = (b) absolute minimum of y = occurring at x =

Explanation:

Step1: Find the derivative

The derivative of $f(x)=x - 5\ln(x)$ is $f^\prime(x)=1-\frac{5}{x}=\frac{x - 5}{x}$.

Step2: Find critical points

Set $f^\prime(x)=0$, so $\frac{x - 5}{x}=0$. The numerator must be 0, giving $x = 5$. Also, $f^\prime(x)$ is undefined at $x=0$, but $0$ is not in the interval $[\frac{1}{5},10]$, so we only consider $x = 5$.

Step3: Evaluate the function at critical - point and endpoints

Evaluate $f(x)$ at $x=\frac{1}{5},5,10$.
$f(\frac{1}{5})=\frac{1}{5}-5\ln(\frac{1}{5})=\frac{1}{5}+5\ln(5)\approx\frac{1}{5}+5\times1.6094 = 0.2+8.047=8.247$.
$f(5)=5 - 5\ln(5)=5(1-\ln(5))\approx5(1 - 1.6094)=5\times(- 0.6094)=-3.047$.
$f(10)=10 - 5\ln(10)=10-5\times2.3026=10 - 11.513=-1.513$.

Answer:

(a) Absolute Maximum of $y = 8.2470$ occurring at $x=\frac{1}{5}$
(b) Absolute Minimum of $y=-3.0470$ occurring at $x = 5$