Use a sign chart to determine the intervals on which f(x)=4x^3-x^4 is concave up and concave down, and identify the locations of any inflection points.
To use a sign chart to determine the intervals of concavity and locate the inflection points of a function, you would need to follow these steps:
Step 1: Find the second derivative of the given function, f(x), which will give you the concavity of the function. In this case, let's find the second derivative of f(x) = 4x^3 - x^4.
Taking the derivative, we obtain:
f'(x) = 12x^2 - 4x^3
Taking the derivative again, we obtain:
f''(x) = 24x - 12x^2
Step 2: Create a sign chart by choosing test values in each interval of the function. These test values will help determine the sign of the second derivative in each interval.
Let's mark the critical points for f''(x) = 24x - 12x^2. These occur when f''(x) = 0:
24x - 12x^2 = 0
12x(2 - x) = 0
So, we have two critical points: x = 0 and x = 2.
Step 3: Select test values for each of the intervals created by the critical points. The intervals to consider will be (-∞, 0), (0, 2), and (2, ∞). Choosing test values within each interval, plug them into the second derivative, f''(x), and determine the sign:
For interval (-∞, 0):
Choosing x = -1, we have f''(-1) = 24(-1) - 12(-1)^2 = -12. (Negative sign)
For interval (0, 2):
Choosing x = 1, we have f''(1) = 24(1) - 12(1)^2 = 12. (Positive sign)
For interval (2, ∞):
Choosing x = 3, we have f''(3) = 24(3) - 12(3)^2 = -36. (Negative sign)
Step 4: Finally, use the sign chart to determine the intervals of concavity and the locations of inflection points.
From the sign chart, we observe:
- On the interval (-∞, 0), f''(x) is negative, indicating concave down.
- On the interval (0, 2), f''(x) is positive, indicating concave up.
- On the interval (2, ∞), f''(x) is negative, indicating concave down.
Since the sign of f''(x) changes from concave down to concave up at x = 0 and from concave up to concave down at x = 2, we have two inflection points at x = 0 and x = 2.
Therefore, using the sign chart, we determined the intervals of concavity (concave up or concave down) and identified the locations of the inflection points of the function f(x) = 4x^3 - x^4.