find the coordinates of all of the points on the graph of ( y = f(x) ) that have horizontal tangents.\n\n(…

find the coordinates of all of the points on the graph of ( y = f(x) ) that have horizontal tangents.\n\n( f(x)=14 x - x^{2} )\n\nselect the correct choice below and, if necessary, fill in the answer box to complete your choice.\n\na. the point(s) is/are\n(type an ordered pair, using integers or decimals. use a comma to separate answers as needed.)\nb. there are no points that have horizontal tangents.

find the coordinates of all of the points on the graph of ( y = f(x) ) that have horizontal tangents.\n\n( f(x)=14 x - x^{2} )\n\nselect the correct choice below and, if necessary, fill in the answer box to complete your choice.\n\na. the point(s) is/are\n(type an ordered pair, using integers or decimals. use a comma to separate answers as needed.)\nb. there are no points that have horizontal tangents.

Answer

Explanation:

Step1: Find the derivative of (f(x))

The derivative of (f(x)=14x - x^{2}) using the power rule ((x^{n})^\prime=nx^{n - 1}) is (f^\prime(x)=14-2x).

Step2: Set the derivative equal to zero

A horizontal tangent occurs when (f^\prime(x) = 0). So we set (14-2x=0). Solving for (x): [ \begin{align*} 14-2x&=0\ -2x&=-14\ x& = 7 \end{align*} ]

Step3: Find the (y) - value

Substitute (x = 7) into the original function (f(x)=14x - x^{2}). Then (y=f(7)=14\times7-7^{2}=98 - 49=49).

Answer:

A. The point(s) is/are ((7,49))