consider the following function on the given ...
consider the following function on the given interval. f(x)=19 + 4x - x², 0, 5 find the derivative of the function. f(x)=4 - 2x find any critical numbers of the function. (enter your answers as a comma - separated list. if an answer does not exist, enter dne.) x = 2 find the absolute maximum and absolute minimum values of f on the given interval. absolute minimum value absolute maximum value need help? read it
Answer
# Explanation:
## Step1: Evaluate function at critical - point
We know the critical number is $x = 2$. Substitute $x = 2$ into $f(x)=19 + 4x-x^{2}$.
$f(2)=19+4\times2 - 2^{2}=19 + 8-4=23$.
## Step2: Evaluate function at endpoints
For the left - hand endpoint $x = 0$ of the interval $[0,5]$, substitute into $f(x)$:
$f(0)=19+4\times0 - 0^{2}=19$.
For the right - hand endpoint $x = 5$ of the interval $[0,5]$, substitute into $f(x)$:
$f(5)=19+4\times5 - 5^{2}=19 + 20-25=14$.
## Step3: Compare values
We have $f(0)=19$, $f(2)=23$, and $f(5)=14$.
# Answer:
absolute minimum value: $14$
absolute maximum value: $23$