Use calculus and algebraic methods to do a complete analysis (i.e., intervals of increase and decrease, intercepts, critical points, points of inflection, and intervals of concavity) for each of the following functions and then sketch a graph of the function. (19 marks)
ƒ(x) = x3 - 3x2
f(x) = x^3 - 3x^2
= x^2(x-3)
You should be familiar with the general shape of
y = 1x^3 + .....
yours crosses at x = 3 and since you have a double root at x = 0, it touches the x-axis at x = 0
find f'(x) and f''(x) , that should be easy
when f'(x) is positive, your function increases
when f'(x) is negative, your function decreases
when f'(x) = 0 , you have "critical" points
solve f'(x) = 0 , you get the x's for those points, sub the x's back into f(x) to find the y of those points.
for points of inflection, set f''(x) = 0, find the x of the point, go back to f(x) to find the y
If f''(x) is positive, the function is concave up
if f''(x) is negative, the function is concave down
this is a pretty straightforward question, but is a good one to show all those properties
f = x^3-3x^2 = x^2(x-3)
f' = 3x^2-6x = 3x(x-2)
f" = 6x-6 = 6(x-1)
Now just apply the features you know:
f=0: x-intercepts
x=0: y-intercept
f'=0 ==> critical point
f"=0 ==> inflection
f'>0 ==> increasing
f">0 ==> concave up
To perform a complete analysis of the function ƒ(x) = x^3 - 3x^2, we need to find its intervals of increase and decrease, intercepts, critical points, points of inflection, and intervals of concavity.
1. Intercepts:
To find the x-intercepts, we set ƒ(x) equal to zero and solve for x:
0 = x^3 - 3x^2
Factorizing as much as possible, we get:
0 = x^2(x - 3)
This equation is satisfied when either x^2 = 0 or (x - 3) = 0.
Solving these equations, we find two distinct x-intercepts: x = 0 and x = 3.
To find the y-intercept, we substitute x = 0 into the function:
ƒ(0) = 0^3 - 3(0)^2 = 0
So the y-intercept is y = 0.
2. Intervals of Increase and Decrease:
To determine the intervals of increase and decrease, we need to find the critical points. These occur when the derivative of the function is equal to zero or does not exist.
First, let's find the derivative of ƒ(x):
ƒ'(x) = d/dx (x^3 - 3x^2)
= 3x^2 - 6x
Next, we set the derivative equal to zero and solve for x:
0 = 3x^2 - 6x
0 = x(3x - 6)
This equation is satisfied when either x = 0 or 3x - 6 = 0.
Solving for x, we find two critical points: x = 0 and x = 2.
Now, we can create a sign chart to determine the intervals of increase and decrease:
-∞ 0 2 +∞
-------|-----|-----|------
ƒ'(x) | + | - | +
From the sign chart, we see that ƒ'(x) is positive to the left of x = 0, which means ƒ(x) is increasing on that interval. Similarly, ƒ'(x) is negative between x = 0 and x = 2, so ƒ(x) is decreasing on that interval. Finally, ƒ'(x) is positive to the right of x = 2, indicating that ƒ(x) is increasing again.
3. Points of Inflection:
To find the points of inflection, we need to determine where the concavity changes. This occurs at x-values where the second derivative of the function is equal to zero or does not exist.
Let's find the second derivative of ƒ(x):
ƒ''(x) = d/dx (3x^2 - 6x)
= 6x - 6
Setting the second derivative equal to zero and solving for x, we get:
0 = 6x - 6
6x = 6
x = 1
Hence, the point of inflection is x = 1.
4. Intervals of Concavity:
To determine the intervals of concavity, we can create another sign chart using the second derivative:
-∞ 1 +∞
------|-----|------
ƒ''(x) | - | +
From the sign chart, we see that ƒ''(x) is negative for all x < 1, indicating concavity down. Beyond x = 1, ƒ''(x) is positive, indicating concavity up.
Drawing a summary, we have the complete analysis of the function ƒ(x) = x^3 - 3x^2:
- x-intercepts: (0, 0), (3, 0)
- y-intercept: (0, 0)
- Intervals of increase: (-∞, 0) and (2, +∞)
- Intervals of decrease: (0, 2)
- Critical points: x = 0, x = 2
- Points of inflection: x = 1
- Intervals of concavity: (-∞, 1) and (1, +∞)
Now, to sketch the graph of the function, you can plot these points and use the information gathered from the analysis to connect the dots and determine the shape of the curve. Remember to label the intercepts, critical points, and points of inflection on the graph.