If f(x) is a continuous function with f"(x)=-5x^2(2x-1)^2(3x+1)^3 , find the set of values for x for which f(x) has an inflection point.
A. {0,-1/3,1/2}
B. {-1/3,1/2}
C. {-1/3}
D. {1/2}
E. No inflection points
clearly, f"=0 at {0,1/2,-1/3}
However, for an inflection point, f' must not change sign. (Think of the graph of x^3)
The function for f'(x) is a nasty polynomial, so I assume that (A) is correct. If you actually do the calculations, you will see that f'(x) > 0 for the points in question, so it does not change sign.
To find the values of x for which f(x) has an inflection point, we need to analyze the concavity of the function.
An inflection point occurs where the concavity of the function changes. This happens when the second derivative changes sign.
From the given information, we have f''(x) = -5x^2(2x-1)^2(3x+1)^3.
To find the values of x where f''(x) changes sign, we need to determine where each factor changes sign.
Let's analyze each factor individually:
1. x^2 changes sign at x = 0.
2. (2x-1)^2 changes sign at x = 1/2.
3. (3x+1)^3 changes sign at x = -1/3.
Therefore, f''(x) changes sign at x = 0, x = 1/2, and x = -1/3.
To determine the concavity of f(x) at these points, we can consider the sign of f''(x) on both sides of each point. Here's the concavity analysis:
1. At x < -1/3, f''(x) < 0, so the concavity of f(x) is down (concave down).
2. Between -1/3 and 0, f''(x) > 0, so the concavity of f(x) is up (concave up).
3. Between 0 and 1/2, f''(x) < 0, so the concavity of f(x) is down (concave down).
4. At x > 1/2, f''(x) > 0, so the concavity of f(x) is up (concave up).
Based on the concavity analysis, f(x) has an inflection point at x = -1/3, where the concavity changes from down to up.
Therefore, the set of values for x for which f(x) has an inflection point is C. {-1/3}.
To find the set of values for x where the function f(x) has an inflection point, we need to analyze the concavity of the function.
In order to determine the concavity at any given point, we need to consider the second derivative of the function f(x).
Given that f"(x) = -5x^2(2x-1)^2(3x+1)^3, we can see that the sign of the second derivative changes when the factors of the function change sign. In other words, the concavity changes when any of the factors change sign.
Let's analyze each factor individually:
1. x^2: This factor is always positive or zero, so it does not contribute to any changes in concavity.
2. (2x-1)^2: This factor changes sign at x = 1/2 since (2x-1) = 0 at x = 1/2.
3. (3x+1)^3: This factor changes sign at x = -1/3 since (3x+1) = 0 at x = -1/3.
Therefore, the set of values for x where the function f(x) has an inflection point is the set of values where the factors change sign, which is at x = -1/3 and x = 1/2.
Hence, the correct answer is option B, {-1/3, 1/2}.