3. Determine the maximum and minimum value coordinates of the graph of the curve:
y = x3/3 – x2/2 – 6x + 5/3
since the curve extends forever in both directions, there is no max or min.
However, the local max/min occur when
y'=0
That is, when
x^2-x-6 = 0
Now finish it off.
To determine the maximum and minimum value coordinates of the graph of the curve, we need to find the critical points of the function. These are the points where the derivative is either zero or undefined.
First, let's find the derivative of the function by applying the power rule to each term:
dy/dx = (1/3)(3x^2) - (1/2)(2x) - 6
Simplifying the derivative, we get:
dy/dx = x^2 - x - 6
Next, we set the derivative equal to zero and solve for x to find the critical points:
x^2 - x - 6 = 0
Factoring the quadratic equation, we have:
(x - 3)(x + 2) = 0
Setting each factor equal to zero gives us two possible critical points:
x - 3 = 0 --> x = 3
x + 2 = 0 --> x = -2
Now, we need to determine if these points are maximum or minimum points. To do this, we can analyze the second derivative of the function. If the second derivative is positive, the point is a minimum; if the second derivative is negative, the point is a maximum.
To find the second derivative, we differentiate the derivative function:
d^2y/dx^2 = 2x - 1
Substituting the critical points into the second derivative:
For x = 3: d^2y/dx^2 = 2(3) - 1 = 5 (positive, so a minimum)
For x = -2: d^2y/dx^2 = 2(-2) - 1 = -5 (negative, so a maximum)
Therefore, the maximum value coordinate is (-2, f(-2)) and the minimum value coordinate is (3, f(3)), where f(x) is the given function.
To find the corresponding y-coordinates, we substitute the x-values into the original function:
For x = -2:
y = (-2)^3/3 - (-2)^2/2 - 6(-2) + 5/3
y = -8/3 - 2 - (-12) + 5/3
y = -8/3 - 2 + 12 + 5/3
y = -8/3 + 6 + 5/3
y = 3/3 (or 1)
So, the maximum value coordinate is (-2, 1).
For x = 3:
y = (3)^3/3 - (3)^2/2 - 6(3) + 5/3
y = 27/3 - 9/2 - 18 + 5/3
y = 9 - 4.5 - 18 + 5/3
y = 4.5 - 22.5 + 5/3
y = -13.5 + 1.66666667
y = -11.83333333
So, the minimum value coordinate is (3, -11.83333333).
Thus, the maximum value coordinates of the graph of the curve are (-2, 1), and the minimum value coordinates are (3, -11.83333333).