Question

Find the points on the curve y = x^3 + 3x^2 − 9x + 3 where the tangent is horizontal.

smaller x-value=____
larger x-value=______

Answer 1

To find the points on the curve where the tangent is horizontal, we need to find the derivative of the curve and set it equal to zero. The derivative of y = x^3 + 3x^2 - 9x + 3 is y' = 3x^2 + 6x - 9. By setting y' equal to zero and solving for x, we can find the two points on the curve.

Step-by-step explanation:

To find the points on the curve where the tangent is horizontal, we need to find the derivative of the curve and set it equal to zero. The derivative of y = x^3 + 3x^2 - 9x + 3 is y' = 3x^2 + 6x - 9. Now, set y' equal to zero and solve for x:

3x^2 + 6x - 9 = 0

Using the quadratic formula, we can find the values of x:

x = (-6 ± √(6^2 - 4(3)(-9))) / (2(3))

Simplifying the equation will give us the two values of x:

Smaller x-value = -2

Larger x-value = 1