Answer :
The function f(x) = 12x⁵ + 45x⁴ − 360x³ + 2 has three inflection points. They are at x = -6, x = 3, and another point between these two values where the concavity changes.
The function f(x) = 12x⁵ + 45x⁴ − 360x³ + 2 has three inflection points. An inflection point is a point on the graph where the concavity changes. In this case, it means the points where the graph changes from concave up to concave down or vice versa. To find the inflection points, we need to find the values of x where the second derivative of f(x) changes sign.
To determine the inflection points, we first calculate the second derivative of the function f(x). The second derivative is the derivative of the first derivative of f(x). Let's call the second derivative g(x).
g(x) = d²f(x)/dx² = d/dx (df(x)/dx)
Now, we take the second derivative of f(x) = 12x⁵ + 45x⁴ − 360x³ + 2:
g(x) = d/dx (12x⁵ + 45x⁴ − 360x³ + 2)
g(x) = 60x⁴ + 180x³ - 1080x²
Next, we set g(x) equal to zero and solve for x to find the critical points:
60x⁴ + 180x³ - 1080x² = 0
Factor out x²:
60x² (x² + 3x - 18) = 0
Now set each factor equal to zero:
60x² = 0 or (x² + 3x - 18) = 0
Solve for x using the quadratic formula:
x = -3 ± √(3² - 4(1)(-18))/(2(1))
x = -3 ± √(9 + 72)/2
x = -3 ± √81/2
x = -3 ± 9/2
x = -12/2 or x = 6/2
x = -6 or x = 3
So the critical points are x = -6 and x = 3. We need to test these points to determine if they are inflection points. We evaluate the second derivative at these points:
g(-6) = 60(-6)⁴ + 180(-6)³ - 1080(-6)²
= 0
g(3) = 60(3)⁴ + 180(3)³ - 1080(3)²
= 1080
Since the second derivative changes sign at x = 3, it is an inflection point. Therefore, the three inflection points of f(x) = 12x⁵ + 45x⁴ − 360x³ + 2 are x = -6, x = 3, and another point between these two values where the concavity changes.
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