Question

F(x)=x

4
−3x
3
+8 Demermine the intervals on which the given Aunction is concave up or concave down. Select the cortect choice beiow and tat in the answer boxies) to complete your choice. (Simplify your antwer. Type your answer in interval notation. Use integens or fractions for any numbers in the expression. Use a comma to separale answers as needed) A. The function is concave down on B. The function is concive up on c. The function h concave up on and concave down on

Answer 1

The correct option c.

To determine the intervals where the given function
`\( F(x) = x^4 - 3x^3 + 8 \)` is concave up or concave down, we need to find the second derivative and analyze its sign.

1. Find the first derivative
`\( F'(x) \)`:

`\[ F'(x) = 4x^3 - 9x^2 \]`

2. Find the second derivative
`\( F''(x) \)`:

`\[ F''(x) = 12x^2 - 18x \]`

Now, set
`\( F''(x) \)` equal to zero and solve for x:

`\[ 12x^2 - 18x = 0 \]`

Factor out
`\( 6x \)`:

`\[ 6x(2x - 3) = 0 \]`

This gives
`\( x = 0 \) and \( x = (3)/(2) \)` as critical points.

Now, test the intervals created by these critical points using the second derivative test:

- For
`\( x < 0 \)`, choose
`\( x = -1 \)` (test point).

`\[ F''(-1) = 12(-1)^2 - 18(-1) = 30 > 0 \]`

The function is concave up on
`\( (-\infty, 0) \)`.

- For
`\( 0 < x < (3)/(2) \)`, choose
`\( x = 1 \)` (test point).

`\[ F''(1) = 12(1)^2 - 18(1) = -6 < 0 \]`

The function is concave down on
`\( (0, (3)/(2)) \)`.

- For
`\( x > (3)/(2) \)`, choose
`\( x = 2 \)` (test point).

`\[ F''(2) = 12(2)^2 - 18(2) = 12 > 0 \]`

The function is concave up on
`\( ((3)/(2), \infty) \)`.

Therefore:

- The function is concave up on
`\( (-\infty, 0) \cup ((3)/(2), \infty) \)`

- The function is concave down on
`\( (0, (3)/(2)) \)`.

In interval notation:

A. The function is concave down on
`(0, (3)/(2)) \]`

B. The function is concave up on
`(-\infty, 0) \cup ((3)/(2), \infty) \]`

C. The function is concave up on
`(-\infty, 0) \cup ((3)/(2), \infty) \text{ and concave down on } (0, (3)/(2)) \]`