Question

Use the derivative of f(x) = 6x⁷ + 7x⁶ to determine any points on the graph of f(x) at which the tangent line is horizontal

a. f(x) has no points with a horizontal tangent line
b. (0,0) and (-1,1)
c. (1, 13) and (-1,1)
d. (-1, -24)
e. (6,0)

Answer 1

To find the points on the graph of f(x) with horizontal tangent lines, we set the derivative equal to zero and solve for x. The points are (0,0) and (-1,1).

Step-by-step explanation:

To determine the points on the graph of f(x) at which the tangent line is horizontal, we need to find the values of x where the derivative of f(x) equals zero.

The derivative of f(x) = 6x⁷ + 7x⁶ is f'(x) = 42x⁶ + 42x⁵.

Setting f'(x) equal to zero and solving for x, we get:

42x⁶ + 42x⁵ = 0

Factoring out 42x⁵, we have:

42x⁵(x + 1) = 0

So, the values of x where the tangent line is horizontal are x = 0 and x = -1.

Therefore, the correct answer is (0,0) and (-1,1).