A) Determine the location of the turning pints for the cubic y= x^3 - 3x^2 +2x+10
To determine the location of turning points for the cubic function y = x^3 - 3x^2 + 2x + 10, we need to find the points where the derivative of the function equals zero.
Step 1: Find the derivative of the function.
The derivative of y = x^3 - 3x^2 + 2x + 10 can be found by using the power rule. Take the derivative of each term separately:
dy/dx = 3x^2 - 6x + 2.
Step 2: Set the derivative equal to zero and solve for x.
Setting 3x^2 - 6x + 2 = 0, we need to solve this quadratic equation. You can use the quadratic formula to find the solutions:
x = (-b ± √(b^2 - 4ac)) / 2a.
For our quadratic equation, the values are:
a = 3, b = -6, c = 2.
Plugging these values into the quadratic formula, we have:
x = (-(-6) ± √((-6)^2 - 4(3)(2))) / (2(3)).
Simplifying further, we get:
x = (6 ± √(36 - 24)) / 6,
x = (6 ± √12) / 6.
Step 3: Simplify the expressions for x, if possible.
x = (6 ± 2√3) / 6 can be simplified by dividing both numerator and denominator by 2:
x = (3 ± √3) / 3.
So, the x-values for the turning points are x = (3 + √3) / 3 and x = (3 - √3) / 3.
Step 4: Find the corresponding y-values for each x-value calculated above.
To find the y-values at the turning points, substitute the x-values into the original cubic equation.
For x = (3 + √3) / 3, substitute this value into the equation:
y = ((3 + √3) / 3)^3 - 3((3 + √3) / 3)^2 + 2((3 + √3) / 3) + 10.
Calculate and simplify to find the y-value.
For x = (3 - √3) / 3, substitute this value into the equation:
y = ((3 - √3) / 3)^3 - 3((3 - √3) / 3)^2 + 2((3 - √3) / 3) + 10.
Calculate and simplify to find the y-value.
These y-values correspond to the turning points of the cubic function y = x^3 - 3x^2 + 2x + 10.