for the function, find the points on the graph at which the tangent line is horizontal.\n\n( y = x ^ { 2 }…

for the function, find the points on the graph at which the tangent line is horizontal.\n\n( y = x ^ { 2 } - 6 )\n\na. the point(s) at which the tangent line is horizontal is(are) (type an ordered pair. use a comma to separate answers as needed.)\nb. the tangent line is horizontal at all points of the graph.\nc. there are no points on the graph where the tangent line is horizontal.

for the function, find the points on the graph at which the tangent line is horizontal.\n\n( y = x ^ { 2 } - 6 )\n\na. the point(s) at which the tangent line is horizontal is(are) (type an ordered pair. use a comma to separate answers as needed.)\nb. the tangent line is horizontal at all points of the graph.\nc. there are no points on the graph where the tangent line is horizontal.

Answer

Explanation:

Step1: Find the derivative of the function

The derivative of (y = x^{2}-6) using the power rule ((x^{n})^\prime=nx^{n - 1}) is (y^\prime=\frac{d}{dx}(x^{2}-6)=2x).

Step2: Set the derivative equal to zero

Since the slope of a horizontal tangent line is (0), we set (y^\prime = 0). So, (2x=0).

Step3: Solve for (x)

Dividing both sides of (2x = 0) by (2), we get (x = 0).

Step4: Find the corresponding (y)-value

Substitute (x = 0) into the original function (y=x^{2}-6). Then (y=(0)^{2}-6=-6).

Answer:

A. The point(s) at which the tangent line is horizontal is(are) ((0,-6))