Answer :
Final answer:
To find the inflection points of the function f(x) = 4x⁴ + 23x³ - 9x² + 5, we need to find the second derivative of the function and solve for x when the second derivative is equal to zero. The inflection points occur where the concavity of the function changes. The inflection points are approximately x ≈ -2.06 and x ≈ 0.16.
Explanation:
To find the inflection points of the function f(x) = 4x⁴ + 23x³ - 9x² + 5, we need to first find the second derivative of the function. The second derivative will tell us where the concavity of the function changes. The inflection points occur where the concavity changes from concave up to concave down or vice versa.
The second derivative of f(x) is f''(x) = 72x² + 138x - 18. To find the values of x that make the second derivative equal to zero, we set f''(x) = 0 and solve for x.
By solving the equation 72x² + 138x - 18 = 0, we find two values of x: x ≈ -2.06 and x ≈ 0.16. Therefore, the inflection points of the function f(x) = 4x⁴ + 23x³ - 9x² + 5 are approximately x ≈ -2.06 and x ≈ 0.16.
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