For any constant c, define the function f_c(x)= x^3+2x^2+cx.
(a) Graph y = f_c(x) for these values of the parameter c: c = -1, 0, 1, 2, 3, 4. What are the similarities and differences among the graphs, and how do the graphs change as the parameter increases?
(b) For what values of the parameter c will f_c have one local maximum and one local minimum? Use Calculus. As c increases, what happens to the distance between the local maximum and the local minimum?
(c) For what values of the parameter c will f_c have no local maximum or local minimum? Use calculus.
(d) Are there any values of the parameter c for which f_c will have exactly one horizontal tangent line?
When c changes, the slope of cubic function changes? How is it changing?
What are the similarities and differences?
To answer these questions, we'll analyze the function f_c(x) = x^3 + 2x^2 + cx.
(a) To graph y = f_c(x) for different values of c, we can use a graphing tool or software like Desmos or Wolfram Alpha. Start by plugging in the given values of c (-1, 0, 1, 2, 3, 4) into the function. Plot the resulting curves on the same coordinate system.
By observing the graphs, you can identify similarities and differences among them. The main similarity is that all the graphs are cubic functions and share the same shape, characterized by a main curve that rises to the left and right. The differences lie in the specific values of the constant c, which affect the graph's steepness, position, and the number and location of its critical points.
As the parameter c increases, you will notice the following changes:
- The graph becomes steeper or flatter depending on the sign of c.
- The position of the graph shifts either upward or downward along the y-axis.
- The number and location of the critical points (local maximum and minimum) may change.
(b) To find the values of c for which f_c(x) has one local maximum and one local minimum, we need to examine the critical points of the function. A critical point occurs where the derivative of the function is equal to zero or undefined.
Taking the derivative of f_c(x) with respect to x, we get:
f'_c(x) = 3x^2 + 4x + c
For one local maximum and one local minimum, we require f'_c(x) to have two distinct real roots.
To find the discriminant of the quadratic expression 3x^2 + 4x + c, we use the formula:
Discriminant (D) = b^2 - 4ac
For two distinct real roots, D > 0. In this case, we have:
(4^2) - (4 * 3 * c) > 0
16 - 12c > 0
16 > 12c
4/3 > c
So, for f_c(x) to have one local maximum and one local minimum, the parameter c must be less than 4/3.
As the parameter c increases, the distance between the local maximum and the local minimum decreases. This is because as c increases, the graph becomes flatter, resulting in a smaller difference between the maximum and minimum values.
(c) To find the values of c for which f_c(x) has no local maximum or local minimum, we need to consider two cases:
1. When f'_c(x) has no real roots.
2. When f'_c(x) has one root.
For f'_c(x) = 3x^2 + 4x + c to have no real roots, the discriminant D < 0. In this case, we have:
(4^2) - (4 * 3 * c) < 0
16 - 12c < 0
16 < 12c
4/3 < c
So, for f_c(x) to have no local maximum or local minimum, the parameter c must be greater than 4/3.
(d) To find the values of c for which f_c(x) has exactly one horizontal tangent line, we need to find where f'_c(x) is equal to zero but has no real roots.
In other words, we need to find values of c where the quadratic function 3x^2 + 4x + c has a "double root." This occurs when the discriminant D is equal to zero:
(4^2) - (4 * 3 * c) = 0
16 - 12c = 0
16 = 12c
c = 16/12
c = 4/3
Therefore, when c = 4/3, f_c(x) will have exactly one horizontal tangent line.