Can you please help me? I try very hard but I don’t understand how to do
Given the function f (x) = 2x^3 + 24x^2 + 72x, determine all intervals on which f’ is increasing.
Thank you for the answer.
Thank you soooo much.
If you want f'(x) to increase, then f''(x) must be positive
f(x) = 2x^3 + 24x^2 + 72x
f'(x) = 6x^2 + 48x + 72
f''(x) = 12x + 48
so you want 12x + 48 > 0
12x > -48
x > -4
The question could have been asked:
For what values of x is f(x) concave upwards, or opens upwards (holds water)
Look at the original graph, notice you would have a point of inflection when x = -4
I would be concave upwards, (holds water) for x > -4, have a point of inflection at
x = 4 and concave downwards for x < -4
Certainly! I'd be happy to help you with that.
To determine the intervals on which the derivative function (f') is increasing, we need to follow some steps.
Step 1: Find the derivative of the given function, f(x).
The derivative of a function gives us information about its rate of change. In this case, we need to find the derivative of f(x) = 2x^3 + 24x^2 + 72x. Applying the power rule, we can find the derivative as follows:
f'(x) = 6x^2 + 48x + 72
Step 2: Set up the inequality for f'(x) > 0.
To determine the intervals where f' is increasing, we need to find the intervals where the derivative is greater than 0 (positive values).
6x^2 + 48x + 72 > 0
Step 3: Solve the inequality.
To solve this quadratic inequality, we can use different methods like factoring, completing the square, or the quadratic formula. However, in this case, it is easier to solve by factoring:
6(x^2 + 8x + 12) > 0
Now, we need to find the intervals where this expression is greater than 0.
Step 4: Find the critical points.
To find the critical points, we equate the quadratic expression to zero:
x^2 + 8x + 12 = 0
Factoring this quadratic equation, we get:
(x + 2)(x + 6) = 0
x = -2 or x = -6
So, the critical points are x = -2 and x = -6.
Step 5: Plot the intervals on a number line.
Now, we can plot the critical points and choose values within each interval to determine the sign of the expression.
Number line: (-∞, -6) | (-6, -2) | (-2, ∞)
Step 6: Test the intervals.
Choose a test point from each interval:
For (-∞, -6), we can choose x = -7.
Substituting x = -7 into 6(x^2 + 8x + 12) > 0 gives us:
6(-7^2 + 8(-7) + 12) = -42 + 56 + 72 = 86 (greater than zero).
For (-6, -2), we can choose x = -5.
Substituting x = -5 into 6(x^2 + 8x + 12) > 0 gives us:
6(-5^2 + 8(-5) + 12) = -90 (less than zero).
For (-2, ∞), we can choose x = 0.
Substituting x = 0 into 6(x^2 + 8x + 12) > 0 gives us:
6(0^2 + 8(0) + 12) = 72 (greater than zero).
Step 7: Determine the intervals.
Based on the signs obtained from our test points, we can determine the intervals where f' is increasing.
The intervals where f' > 0 (increasing) are: (-∞, -6) and (-2, ∞).
So, the answer to your question is that f' is increasing on the intervals (-∞, -6) and (-2, ∞).