Question

Use the limit definition of slope at a point to find the point on f(x)=x²−6x+2 where the tangent line is horizontal.

a. (3,−9)
b. (6,−18)
c. (2,−4)
d.(0,2)

Answer 1

The point on the function f(x) = x² - 6x + 2 where the tangent is horizontal is found by setting the derivative (2x - 6) to zero, resulting in x = 3. However, there's a typo in the options, as the correct point is (3, -7) not provided in the choices.

Step-by-step explanation:

To find the point on the curve f(x) = x² - 6x + 2 where the tangent line is horizontal, we must locate where the slope of the tangent (the derivative of the function) is zero. The derivative of f(x) is found using the power rule. Specifically:

f'(x) = 2x - 6

Setting this derivative equal to zero to find the x-coordinate where the tangent is horizontal, we get:

2x - 6 = 0
x = 3

Plugging x = 3 back into the original function, we get:

f(3) = 3² - 6(3) + 2
f(3) = 9 - 18 + 2
f(3) = -7

However, there seems to be a typo in the options provided, as none of them include the point (3, -7).