consider the following function on the given interval. f(x)=19 + 4x - x², 0, 5 find the derivative of the…

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

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: $23$