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Answer :
Final answer:
To find the inflection points of the function f(x) = x⁴ + x³-45x² + 7, we need to find the second derivative, set it equal to zero, and solve for x. The inflection points of the function are x = -3 and x = 5/2.
Explanation:
To find the inflection points of the function f(x) = x⁴ + x³-45x² + 7, we need to find the second derivative of the function. First, we find the first derivative of f(x): f'(x) = 4x³ + 3x² - 90x. Then, we find the second derivative: f''(x) = 12x² + 6x - 90.
Next, we set the second derivative equal to zero and solve for x: 12x² + 6x - 90 = 0.
Using the quadratic formula, x = (-b ± √(b² - 4ac)) / (2a)
In our equation, a = 12, b = 6, and c = -90.
Therefore,x = (-6 ± √(6² - 4 * 12 * (-90))) / (2 * 12)
x = (-6 ± √4356) / 24
This implies x = 5/2 and x = -3
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