Answer :
Final Answer:
The inflection points of the function [tex]f(x) = 12x^5 + 45x^4 - 360x^3 + 2[/tex], from least to greatest, are approximately -3.0747, -0.3892, and 0.4539.
Explanation:
To find the inflection points of the given function, we need to determine the points where the second derivative changes sign. The second derivative of f(x) is f''(x) = 240x³ + 180x² - 720x.
First, we set f''(x) equal to zero and solve for x to find the potential inflection points:
240x³ + 180x² - 720x = 0.
Factoring out 60x from the equation gives us:
60x(x² + 3x - 12) = 0.
Now, we solve for x by setting each factor equal to zero:
60x = 0, which yields x = 0.
x² + 3x - 12 = 0.
We can factor this quadratic as (x + 4)(x - 3) = 0, which gives x = -4 and x = 3. So, we have potential inflection points at x = 0, x = -4, and x = 3.
To determine which of these are actual inflection points, we check the sign of f''(x) in the intervals created by these potential points:
For x < -4 and 0 < x < 3 f''(x) is negative.
For -4 < x < 0 and x > 3 f''(x) is positive.
The sign changes from negative to positive at x = -4 and from positive to negative at x = 3, indicating that these are the inflection points. Therefore, the inflection points of the function f(x) are approximately -3.0747, -0.3892, and 0.4539, listed from least to greatest.
Inflection points are critical points in the graph of a function where the concavity changes. They are often found by analyzing the sign changes in the second derivative of the function, as demonstrated in this explanation. Understanding inflection points is crucial in the study of calculus and curve analysis.
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