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Consider the function \( f(x) = 6x + 2x - 1 \).

For this function, there are four important intervals: \((-∞, A]\), \((A, B)\), \((B, C]\), and \([C, ∞)\), where \( A \) and \( C \) are the critical numbers, and the function is not defined at \( B \).

1. Find \( A \), \( B \), and \( C \).
2. For each of the following open intervals, tell whether \( f(x) \) is increasing or decreasing:
- \((-∞, A)\)
- \((A, B)\)
- \((B, C)\)
- \((C, ∞)\)

Note that this function has no inflection points, but we can still consider its concavity.

3. For each of the following intervals, tell whether \( f(x) \) is concave up or concave down:
- \((-∞, B)\)
- \((B, ∞)\)

Consider the function \( f(x) = 12x^5 + 60x^4 - 100x^3 + 1 \).

The function \( f(x) \) has inflection points at (reading from left to right) \( x = D, E, \) and \( F \).

4. For each of the following intervals, tell whether \( f(x) \) is concave up or concave down:
- \((-∞, D)\)
- \((D, E)\)
- \((E, F)\)
- \((F, ∞)\)

Answer :

The concavity of f(x) is as follows:(−∞, D): Concave up(D, E): Concave down(E, ∞): Concave up

To find the critical numbers and points of discontinuity for the function f(x) = 6x + 2x^(-1), let's first find the derivative and identify where it is undefined.

Find the derivative of f(x):

f'(x) = 6 - 2x^(-2)

Identify points of discontinuity:

The function is not defined when x = 0 since x^(-1) is undefined at x = 0.

Find critical numbers:

To find the critical numbers, we set the derivative equal to zero and solve for x:

6 - 2x^(-2) = 0

2x^(-2) = 6

x^(-2) = 3/2

Taking the reciprocal of both sides:

x^2 = 2/3

x = ±√(2/3)

So, the critical numbers are A = -√(2/3) and C = √(2/3). The function is not defined at B = 0.

Now let's determine whether f(x) is increasing or decreasing and its concavity in each of the given intervals:

(−∞, A):

In this interval, f(x) is decreasing since f'(x) = 6 - 2x^(-2) > 0 for x < A.

(A, B):

In this interval, f(x) is increasing since f'(x) = 6 - 2x^(-2) > 0 for A < x < B.

(B, C):

The function is not defined in this interval.

(C, ∞):In this interval, f(x) is increasing since f'(x) = 6 - 2x^(-2) > 0 for x > C.

Next, let's consider the function f(x) = 12x^5 + 60x^4 - 100x^3 + 1:

To find the inflection points, we need to find where the concavity changes. This occurs when the second derivative is zero or undefined.

Find the second derivative of f(x):

f''(x) = 60x^4 + 240x^3 - 300x^2

Find the points of inflection:

Set f''(x) = 0 and solve for x:

60x^4 + 240x^3 - 300x^2 = 0

60x^2(x^2 + 4x - 5) = 0

x^2 + 4x - 5 = 0

Solving this quadratic equation, we find two solutions: x = D and x = E, where D < E.

Determine concavity:

(−∞, D):

In this interval, f(x) is concave up since f''(x) = 60x^4 + 240x^3 - 300x^2 > 0 for x < D.

(D, E):

In this interval, f(x) is concave down since f''(x) = 60x^4 + 240x^3 - 300x^2 < 0 for D < x < E.

(E, ∞):

In this interval, f(x) is concave up since f''(x) = 60x^4 + 240x^3 - 300x^2 > 0 for x > E.

Therefore, the concavity of f(x) is as follows:

(−∞, D): Concave up

(D, E): Concave down

(E, ∞): Concave up

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