a.)Determine the location of the turning points for the cubic y=x^3-3x^2+2x+10. What are the maximum and minimum values for this function?
b.)Use the fact that a cubic is symmetric about its point of inflection to determine the location of the point of inflection of the cubic in part (a).
pre cal...wondering if you have had derivites.
y'=2x^2-6x+2=0
so max, min occur at (x-1)(x-2)=0 or at x=1, or x=2
y"=4x-6=0 or x=1.5 (halfway between roots at 1 and 2)
values of the max..putx=1, x=2 into the original equation, and commpute y.
so what are the turning points
To determine the location of the turning points for the cubic function y= x^3 - 3x^2 + 2x + 10, we can find the critical points by taking the derivative of the function and setting it equal to zero.
a.) Finding the turning points:
Step 1: Take the derivative of the cubic function.
The derivative of y = x^3 - 3x^2 + 2x + 10 is:
dy/dx = 3x^2 - 6x + 2
Step 2: Set the derivative equal to zero and solve for x.
0 = 3x^2 - 6x + 2
Step 3: Use the quadratic formula to solve for x:
x = (-b ± √(b^2 - 4ac))/(2a)
In this case, a = 3, b = -6, and c = 2. Plugging in these values, we can solve for x.
x = (-(-6) ± √((-6)^2 - 4(3)(2)))/(2(3))
x = (6 ± √(36 - 24))/(6)
x = (6 ± √12)/(6)
x = (6 ± 2√3)/(6)
x = 1 ± √3/3
Step 4: Find the corresponding y-values for the critical points.
Plug the values of x into the original cubic function:
y = (1 + √3/3)^3 - 3(1 + √3/3)^2 + 2(1 + √3/3) + 10
y = (1 - √3/3)^3 - 3(1 - √3/3)^2 + 2(1 - √3/3) + 10
These give the x and y coordinates of the critical points, which are the turning points of the cubic function.
To find the maximum and minimum values for this cubic function, we need to examine the nature of the turning points.
b.) Determining the maximum and minimum values:
To find the maximum and minimum values of the cubic function, we can either use the second derivative test or examine the behavior of the function as x approaches infinity or negative infinity.
For the second derivative test, we need to find the second derivative of the cubic function.
Step 1: Take the derivative of the derivative obtained earlier:
d2y/dx2 = 6x - 6
Step 2: Substitute the x-values of the critical points into the second derivative.
For x = 1 + √3/3:
d2y/dx2 = 6(1 + √3/3) - 6 = 2√3
For x = 1 - √3/3:
d2y/dx2 = 6(1 - √3/3) - 6 = -2√3
Step 3: Analyze the second derivative:
Since the second derivative at x = 1 + √3/3 is positive (2√3) and at x = 1 - √3/3 is negative (-2√3), we can conclude that x = 1 + √3/3 corresponds to a local minimum, and x = 1 - √3/3 corresponds to a local maximum.
Step 4: Calculate the corresponding y-values for the minimum and maximum points.
Plug the values of x into the original cubic equation:
For x = 1 + √3/3:
y = (1 + √3/3)^3 - 3(1 + √3/3)^2 + 2(1 + √3/3) + 10
For x = 1 - √3/3:
y = (1 - √3/3)^3 - 3(1 - √3/3)^2 + 2(1 - √3/3) + 10
These will give you the maximum and minimum values of the cubic function.
Regarding part (b), the statement that a cubic is symmetric about its point of inflection is not always true. The "point of inflection" is where the curvature of the graph changes, and it does not necessarily imply symmetry. However, for this specific cubic function, we can use the symmetry property to determine its point of inflection.
Since a cubic function is symmetric about its point of inflection, the point of inflection lies at the average of the x-values of the minimum and maximum points.
The x-coordinate of the point of inflection is:
(x_max + x_min) / 2
Calculate the average of the x-values obtained earlier and find the corresponding y-value by plugging it into the original cubic equation to determine the location of the point of inflection.