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Answer :
Final answer:
To find the inflection points of the function f(x) = x⁴ - x³ - 45x² + 8, we need to find the second derivative, set it equal to zero to find potential inflection points, and then confirm these points by analyzing the change in concavity around them.
Explanation:
The question asks to find the inflection points of the function f(x) = x⁴ - x³ - 45x² + 8. To do this, we need to perform the following steps:
- Find the second derivative of the function, f''(x).
- Set the second derivative equal to zero and solve for x to find potential inflection points.
- Test the intervals around the solutions to confirm whether the concavity changes, which signifies an inflection point.
Let's begin with the first derivative of f(x):
f'(x) = 4x³ - 3x² - 90x
Now, we calculate the second derivative:
f''(x) = 12x² - 6x - 90
Set f''(x) to zero and solve for x:
0 = 12x² - 6x - 90
Finally, we solve the quadratic equation to find the potential inflection points and test intervals to confirm the inflection points.
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