Question

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

( f(x)=14 x - x^{2} )

select the correct choice below and, if necessary, fill in the answer box to complete your choice.

a. the point(s) is/are
(type an ordered pair, using integers or decimals. use a comma to separate answers as needed.)
b. there are no points that have horizontal tangents.

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 \(x\):

$$
LATEXBLOCK0
$$

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 \((7,49)\)