Answer :
Final answer:
The stationary points of the given function are x = 0, x = 5/6, x = 6/5, and x = 1. They are all inflection points.
Explanation:
To find the stationary points of the function f(x) = 5x⁶ − 36x⁵ + (165/2)x⁴ − 60x³ + 36 and determine if they are minimum, maximum, or inflection points, we need to find where the derivative equals zero. Let's first find the derivative of f(x):
f'(x) = 30x⁵ - 180x⁴ + 165x³ - 180x²
To find the stationary points, we set f'(x) equal to zero and solve for x:
30x⁵ - 180x⁴ + 165x³ - 180x² = 0
From here, we can factor the equation:
x²(30x³ - 180x² + 165x - 180) = 0
x²(6x - 5)(5x - 6)(x - 1) = 0
Now we solve for x:
x² = 0 (x = 0)
6x - 5 = 0 (x = 5/6)
5x - 6 = 0 (x = 6/5)
x - 1 = 0 (x = 1)
So the stationary points are x = 0, x = 5/6, x = 6/5, and x = 1. To determine whether they are minimum, maximum, or inflection points, we can use the second derivative test. Let's find the second derivative:
f''(x) = 150x⁴ - 720x³ + 495x² - 360x
Plugging in the stationary points, we get:
f''(0) = 0
f''(5/6) = -855/4
f''(6/5) = -855/4
f''(1) = -225
Since both f''(5/6) and f''(6/5) are negative, x = 5/6 and x = 6/5 are inflection points. And since f''(1) is also negative, x = 1 is an inflection point as well. x = 0 does not satisfy the second derivative test, so it is inconclusive. Therefore, x = 0 is also an inflection point.
To summarize, the stationary points of the function f(x) = 5x⁶ − 36x⁵ + (165/2)x⁴ − 60x³ + 36 are x = 0, x = 5/6, x = 6/5, and x = 1. They are all inflection points.
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