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Answer :
To find the critical numbers of the function f(x) = 12x^5 + 60x^4 - 100x^3 + 2, we need to find the values of x where the derivative of f(x) is equal to zero or undefined.
Let's start by finding the derivative of f(x):
f'(x) = 60x^4 + 240x^3 - 300x^2
Setting f'(x) equal to zero:
60x^4 + 240x^3 - 300x^2 = 0
Factoring out common terms:
60x^2(x^2 + 4x - 5) = 0
Setting each factor equal to zero:
60x^2 = 0 (gives x = 0)
x^2 + 4x - 5 = 0 (gives two solutions using quadratic formula)
Solving the quadratic equation, we have:
x = (-4 ± √(4^2 - 4(-5))) / 2
x = (-4 ± √(16 + 20)) / 2
x = (-4 ± √36) / 2
x = (-4 ± 6) / 2
The solutions for x are:
x = -5
x = 1
So, the critical numbers are A = -5, B = 0, and C = 1.
Now, to determine the behavior of f(x) at each critical number, we can examine the sign of the derivative f'(x) in the intervals surrounding these critical numbers.
Interval (-∞, A):
For x < -5, the derivative f'(x) = 60x^4 + 240x^3 - 300x^2 > 0. Therefore, f(x) is increasing in this interval.
Interval (A, B):
For -5 < x < 0, the derivative f'(x) = 60x^4 + 240x^3 - 300x^2 < 0. Therefore, f(x) is decreasing in this interval.
Interval (B, C):
For 0 < x < 1, the derivative f'(x) = 60x^4 + 240x^3 - 300x^2 > 0. Therefore, f(x) is increasing in this interval.
Interval (C, ∞):
For x > 1, the derivative f'(x) = 60x^4 + 240x^3 - 300x^2 > 0. Therefore, f(x) is increasing in this interval.
Now, let's determine whether f(x) has a local min, local max, or neither at each critical number.
At A = -5, since f(x) is increasing to the left of A and decreasing to the right of A, f(x) has a local maximum at x = -5.
At B = 0, since f(x) is decreasing to the left of B and increasing to the right of B, f(x) has a local minimum at x = 0.
At C = 1, since f(x) is increasing to the left of C and increasing to the right of C, f(x) does not have a local min or local max at x = 1.
Therefore, the answers are:
A = -5 corresponds to a local maximum (UMAX).
B = 0 corresponds to a local minimum (LMIN).
C = 1 corresponds to neither a local min nor local max (NETHEA).
Learn more about critical numbers here :
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