examine the graph of f(x)=10x^3+3x^2-12x what are the intervals on which the rate of change of the function will be positive
My teacher has not been helping me and I do not understand this question at all any tips will be greatly appreciated
If this is not a calculus class, then about all you can do is look at the graph. You will notice that it has a maximum at about (-0.74,6.47) and a minimum at about (0.54,-4.03).
So, f(x) is decreasing (rate of change is negative) on the interval (-0.74,0.54) and increasing everywhere else. So your answer is
(-∞,-0.74)U(0.54,∞)
If this is a calculus class, then just note that
f'(x) = 6(5x^2+x-2)
f(x) is increasing where f'(x) > 0 on the interval I noted above.
To determine the intervals on which the rate of change of the function is positive, we need to find where the function is increasing.
The rate of change of a function is captured by its derivative. So, to find the intervals where the function is increasing, we'll need to find the derivative of f(x) = 10x^3 + 3x^2 - 12x.
Let's find the derivative of f(x) step-by-step:
Step 1: Differentiate each term separately using the power rule.
The derivative of 10x^3 is: 3 * 10x^(3-1) = 30x^2.
The derivative of 3x^2 is: 2 * 3x^(2-1) = 6x.
The derivative of -12x is: -12.
So, the derivative of f(x) = 10x^3 + 3x^2 - 12x is: f'(x) = 30x^2 + 6x - 12.
Step 2: Set the derivative equal to zero and solve for x to find the critical points.
Setting f'(x) = 0:
30x^2 + 6x - 12 = 0.
Step 3: Solve the quadratic equation for x using factoring, completing the square, or using the quadratic formula.
Factoring:
6(5x^2 + x - 2) = 0.
The quadratic equation 5x^2 + x - 2 can be factored as: (5x - 2)(x + 1) = 0.
Setting each factor equal to zero:
5x - 2 = 0 -> x = 2/5.
x + 1 = 0 -> x = -1.
So, the critical points are x = 2/5 and x = -1.
Step 4: Plot these critical points on a number line and select values within each interval to test the sign of the derivative.
Number line:
(-∞, -1) | (-1, 2/5) | (2/5, ∞)
Now, let's pick a value in each interval and substitute it into the derivative expression f'(x) = 30x^2 + 6x - 12 to test the sign of the derivative in each interval:
For the interval (-∞, -1):
Let's pick x = -2.
Substituting x = -2:
f'(-2) = 30(-2)^2 + 6(-2) - 12
= 120 - 12 - 12
= 96.
Since f'(-2) = 96 > 0, the rate of change is positive in the interval (-∞, -1).
For the interval (-1, 2/5):
Let's pick x = 0.
Substituting x = 0:
f'(0) = 30(0)^2 + 6(0) - 12
= -12.
Since f'(0) = -12 < 0, the rate of change is negative in the interval (-1, 2/5).
For the interval (2/5, ∞):
Let's pick x = 1.
Substituting x = 1:
f'(1) = 30(1)^2 + 6(1) - 12
= 30 + 6 - 12
= 24.
Since f'(1) = 24 > 0, the rate of change is positive in the interval (2/5, ∞).
Therefore, the intervals on which the rate of change of the function f(x) = 10x^3 + 3x^2 - 12x is positive are (-∞, -1) and (2/5, ∞).
To determine the intervals on which the rate of change of the function is positive, we need to analyze the graph of the function and identify where it is increasing. Here's how you can approach it step-by-step:
Step 1: Find the derivative of the function.
The derivative, denoted as f'(x), gives us the rate of change of the function at any given point. In this case, we want to find the derivative of f(x)=10x^3+3x^2-12x.
To find the derivative, we can apply the power rule for differentiation. The power rule states that if f(x) = ax^n, then f'(x) = nax^(n-1).
Therefore, the derivative of f(x) = 10x^3 + 3x^2 - 12x is:
f'(x) = 30x^2 + 6x - 12.
Step 2: Set up the inequality.
To determine the intervals where the rate of change is positive, we need to determine where f'(x) > 0.
Set up the inequality:
f'(x) > 0.
Step 3: Solve the inequality.
To solve the inequality, we need to find the values of x for which f'(x) is greater than zero.
We can solve the inequality by factoring or using quadratic formula, but in this case, we can observe that f'(x) = 30x^2 + 6x - 12 is a quadratic equation that can be factored. Factoring it, we get:
f'(x) = 6(5x^2 + x - 2) = 0.
Setting each factor equal to zero, we solve for x:
5x^2 + x - 2 = 0.
Using factoring or quadratic formula, we find that the solutions to this equation are:
x = -1 and x = 2/5.
Plotting these values on a number line, we have:
-1 | 2/5.
Step 4: Analyze the intervals.
Now that we have the critical points -1 and 2/5, we can analyze the intervals formed on the number line.
For the interval to the left of -1, (−∞, -1), we can select a test point in that interval, say x = -2, and substitute it into the inequality f'(x) > 0.
f'(-2) = 30*(-2)^2 + 6*(-2) - 12 = 108 > 0.
For the interval between -1 and 2/5, (-1, 2/5), we can select another test point within that interval, say x = 0, and substitute it into f'(x) > 0.
f'(0) = 30*(0)^2 + 6*(0) - 12 = -12 < 0.
Lastly, for the interval to the right of 2/5, (2/5, ∞), we can test with x = 1.
f'(1) = 30*(1)^2 + 6*(1) - 12 = 24 > 0.
Based on the test points, we can conclude that the rate of change of f(x) = 10x^3 + 3x^2 - 12x is positive on the intervals: (-∞, -1) and (2/5, ∞).
This means that the function is increasing (positive rate of change) in those intervals.