Question

Find the points on the curve y = x³ - 3x² - 9x + 7 where the tangent is horizontal. What are the coordinates of the point with the smaller x-value (x, y) and the point with the larger x-value (x, y)?

Answer 1

To find the points on the curve where the tangent is horizontal, we need to find the points where the derivative of the curve is equal to zero. The coordinates of the point with the smaller x-value are (-1, 12). The coordinates of the point with the larger x-value are (3, -5).

Step-by-step explanation:

To find the points on the curve where the tangent is horizontal, we need to find the points where the derivative of the curve is equal to zero.

Let's find the derivative of the equation y = x³ - 3x² - 9x + 7 using the power rule:

dy/dx = 3x² - 6x - 9

Setting this derivative equal to zero:

3x² - 6x - 9 = 0

Now, we can solve this quadratic equation to find the x-coordinate of the points where the tangent is horizontal.
We can factor the equation or use the quadratic formula.
After solving, we find two x-values: -1 and 3.

Substituting these x-values back into the original equation, we can find the corresponding y-values.
The coordinates of the point with the smaller x-value are (-1, 12).
The coordinates of the point with the larger x-value are (3, -5).