A Determine the location of the turning pints for the cubic y= x^3 - 3x^2 +2x+10
B) Use the fact that a cubic is symmetric about its point of inflection to determine the location of the pint of inflection of the cubic in part (a)
A) To determine the location of the turning points for the cubic function y = x^3 - 3x^2 + 2x + 10, we need to find the derivative of the function and solve for when the derivative equals zero.
1. Take the derivative of the cubic function:
y' = 3x^2 - 6x + 2
2. Set the derivative equal to zero and solve for x:
3x^2 - 6x + 2 = 0
We can solve this quadratic equation using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
In our case, a = 3, b = -6, and c = 2. Plugging these values into the formula, we get:
x = (6 ± √((-6)^2 - 4 * 3 * 2)) / (2 * 3)
= (6 ± √(36 - 24)) / 6
= (6 ± √12) / 6
Simplifying further:
x = (6 ± 2√3) / 6
= 1 ± √3/3
So, the turning points of the cubic function are located at x = 1 + √3/3 and x = 1 - √3/3.
B) To determine the location of the point of inflection for the cubic function from part (a), we can use the fact that a cubic function is symmetric about its point of inflection.
1. Find the x-coordinate of the point of inflection:
The x-coordinate of the point of inflection is the average of the x-coordinates of the turning points.
Average = (x1 + x2) / 2
= ((1 + √3/3) + (1 - √3/3)) / 2
= 2/3
So, the x-coordinate of the point of inflection is 2/3.
2. Find the y-coordinate of the point of inflection:
Substitute the x-coordinate into the original cubic function:
y = (2/3)^3 - 3(2/3)^2 + 2(2/3) + 10
= 8/27 - 12/9 + 4/3 + 10
= 46/27
So, the point of inflection for the cubic function is located at x = 2/3, y = 46/27.