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?
If f(x)=x^3+2x^2+cx.
Let f'(x)=3x^2+4x+c be the derivative
b. one minimum and one maximum
To have one maximum and one minimum, f'(x)=0 must have distinct roots. For this to happen, the discriminant, Δ, of the quadratic formula must be >0, or b²-4ac > 0. Note: the c in this expression is the constant term of f'(x), which also happens to be c.
c. no min. no max.
In order to have no local maximum and local minimum, then f'(x)=0 should have no real roots. The quadratic will have two complex roots if Δ is less than zero, or b²-4ac < 0.
d. exactly one horizontal line
This will happen when there are two coincident roots. The two roots will be merged together when Δ=0, or b²-4ac=0.
Solve for the value of c (the constant).
At for the value of c where Δ=0, find the value of x0 where f'(x0)=0.
Confirm that f"(x0)=0 which implies a point of inflexion. I got x0=-2/3.
So what does the derivative of the f(x) have to do with the graphs? Or is the derivative part of the main question: define the function f with the formula? And what would be the similarities and differences of the graphs? And if you increase c then does the slope change?
a.
The question want you to
1. graph the function f(x) (NOT f'(x)) for the given values of the parameter.
2. Describe the similarities and differences of the graphs.
3. Describe the effect of the value of c with respect to the appearances of the graphs (of f(x)).
The derivatives part comes later in b,c and d.
Do you have a graphing calculator?
What kind of similarities and differences is it asking for?
If you have graphed the function for the different values of c requested, you will see the differences.
In fact, part a is to help you understand the questions in parts b, c and d.
If you have not graphed the function as requested, you can also read up parts b, c, and d for the different possible cases when c is varied.
So then there are no similarities? And the differences are the shapes and the distinct roots, max and min?
I would expect similarities for larger values of c. The discriminant equals zero at c=4/3. So I expect similarities when c>2. For c≤1, I expect changes in form (max-min, inflexion, etc.)
I do not have a graphing calculator, so I cannot tell you the differences and similarities offhand.
Okay thank you very much
You're welcome!
(a) To graph the function y = f_c(x) for different values of c, we can simply substitute each value of c into the equation and plot the resulting graph.
For c = -1:
y = f_c(x) = x^3 + 2x^2 - x
For c = 0:
y = f_c(x) = x^3 + 2x^2
For c = 1:
y = f_c(x) = x^3 + 2x^2 + x
For c = 2:
y = f_c(x) = x^3 + 2x^2 + 2x
For c = 3:
y = f_c(x) = x^3 + 2x^2 + 3x
For c = 4:
y = f_c(x) = x^3 + 2x^2 + 4x
To observe the similarities and differences among the graphs, we can compare their shapes and key features. All the graphs are cubic functions, and they have similar general shapes. They have a single turning point or local maximum/minimum point. The exact coordinates of these turning points will vary depending on the value of c.
As the parameter c increases, the graphs shift upward along the y-axis. This means that the vertex or turning point of each graph moves higher. Additionally, the overall steepness or slope of the graphs increases as c increases.
(b) To find the values of c for which f_c(x) has one local maximum and one local minimum, we need to analyze the derivatives of the function.
Let's first find the derivative of f_c(x):
f_c'(x) = d/dx (x^3 + 2x^2 + cx)
= 3x^2 + 4x + c
To have one local maximum and one local minimum, the derivative function f_c'(x) should have two distinct roots. In other words, the quadratic equation 3x^2 + 4x + c = 0 should have two real solutions.
To find the conditions for this, we can use the discriminant of the quadratic equation:
Discriminant = b^2 - 4ac
For the quadratic equation 3x^2 + 4x + c = 0, the discriminant should be greater than zero to have two real solutions. So, we have:
(4^2) - (4 * 3 * c) > 0
16 - 12c > 0
-12c > -16
c < 4/3
Therefore, for c values less than 4/3, the function f_c(x) will have one local maximum and one local minimum.
As c increases, the distance between the local maximum and local minimum decreases. This can be observed as the roots of the quadratic equation get closer together.
(c) To find the values of c for which f_c(x) has no local maximum or local minimum, we again need to analyze the derivatives of the function.
If f_c(x) has no local maximum or local minimum, it means that the derivative f_c'(x) has no real solutions or roots. In other words, the quadratic equation 3x^2 + 4x + c = 0 should have either no real solutions or complex solutions.
For a quadratic equation with no real solutions, the discriminant should be negative:
(4^2) - (4 * 3 * c) < 0
16 - 12c < 0
-12c < -16
c > 4/3
Therefore, for c values greater than 4/3, the function f_c(x) will have no local maximum or local minimum.
(d) To find the values of the parameter c for which f_c will have exactly one horizontal tangent line, we need to find the values of c that make the derivative f_c'(x) (or the slope of the tangent line) equal to zero.
Setting the derivative equal to zero:
3x^2 + 4x + c = 0
For a cubic function, there can be at most two solutions for this equation, resulting in two values of x where the derivative equals zero. These values depend on c. Therefore, for no values of c, will f_c have exactly one horizontal tangent line.