find all relative extrema of the function. use the second derivative test where applicable. (if an answer…

find all relative extrema of the function. use the second derivative test where applicable. (if an answer does not exist, enter dne.)\n\n$f(x)=14x - x^{2}$\n\nrelative maximum $(x,y)=$\n\nrelative minimum $(x,y)=$

find all relative extrema of the function. use the second derivative test where applicable. (if an answer does not exist, enter dne.)\n\n$f(x)=14x - x^{2}$\n\nrelative maximum $(x,y)=$\n\nrelative minimum $(x,y)=$

Answer

Explanation:

Step1: Find the first derivative

The function is (f(x)=14x - x^{2}). Using the power rule ((x^n)^\prime=nx^{n - 1}), the first derivative (f^\prime(x)=(14x - x^{2})^\prime=14-2x).

Step2: Find the critical points

Set (f^\prime(x) = 0), so (14-2x=0). Solving for (x): [ \begin{align*} 14-2x&=0\ 2x&=14\ x& = 7 \end{align*} ]

Step3: Find the second derivative

Differentiate (f^\prime(x)=14 - 2x) with respect to (x). Using the power rule, (f^{\prime\prime}(x)=(14-2x)^\prime=- 2).

Step4: Apply the second - derivative test

Since (f^{\prime\prime}(7)=-2<0), the function (f(x)) has a relative maximum at (x = 7). Substitute (x = 7) into the original function (f(x)=14x - x^{2}): (f(7)=14\times7-7^{2}=98 - 49=49).

Since (f^{\prime\prime}(x)=-2) is always negative (there is no value of (x) for which (f^{\prime\prime}(x)>0)), there is no relative minimum.

Answer:

relative maximum ((x,y)=(7,49)) relative minimum ((x,y)=\text{DNE})