Determine the interval(s) over which the graph of f left parenthesis x right parenthesis equals short dash x to the power of 6 minus 6 x to the power of 5 plus 12 x minus 2 is concave up or concave down. Concave up on left parenthesis 0 comma space 4 right parenthesis Concave down on left parenthesis short dash infinity comma space 0 right parenthesis union left parenthesis 4 comma space infinity right parenthesis Concave up on left parenthesis short dash 4 comma space 0 right parenthesis Concave down on left parenthesis short dash infinity comma space short dash 4 right parenthesis union left parenthesis 0 comma space infinity right parenthesis Concave up on left parenthesis short dash infinity comma space 0 right parenthesis union left parenthesis 4 comma space infinity right parenthesis Concave down on left parenthesis 0 comma space 4 right parenthesis Concave up on left parenthesis short dash infinity comma space short dash 4 right parenthesis union left parenthesis 0 comma space infinity right parenthesis Concave down on left parenthesis short dash 4 comma space 0 right parenthesis
Question
Determine the interval(s) over which the graph of f left parenthesis x right parenthesis equals short dash x to the power of 6 minus 6 x to the power of 5 plus 12 x minus 2 is concave up or concave down.
Concave up on left parenthesis 0 comma space 4 right parenthesis Concave down on left parenthesis short dash infinity comma space 0 right parenthesis union left parenthesis 4 comma space infinity right parenthesis
Concave up on left parenthesis short dash 4 comma space 0 right parenthesis Concave down on left parenthesis short dash infinity comma space short dash 4 right parenthesis union left parenthesis 0 comma space infinity right parenthesis
Concave up on left parenthesis short dash infinity comma space 0 right parenthesis union left parenthesis 4 comma space infinity right parenthesis Concave down on left parenthesis 0 comma space 4 right parenthesis
Concave up on left parenthesis short dash infinity comma space short dash 4 right parenthesis union left parenthesis 0 comma space infinity right parenthesis Concave down on left parenthesis short dash 4 comma space 0 right parenthesis
Solution
To determine where the function f(x) = -x^6 + 6x^5 + 12x - 2 is concave up or down, we first need to find its second derivative, as the sign of the second derivative tells us where the function is concave up (second derivative > 0) or concave down (second derivative < 0).
Step 1: Find the first derivative of f(x). f'(x) = -6x^5 + 30x^4 + 12
Step 2: Find the second derivative of f(x). f''(x) = -30x^4 + 120x^3
Step 3: Set the second derivative equal to zero and solve for x to find potential points of inflection. -30x^4 + 120x^3 = 0 x^3(120 - 30x) = 0 x = 0, 4
Step 4: Test the intervals determined by the potential points of inflection in the second derivative.
For x < 0, f''(x) > 0, so the function is concave up on (-∞, 0). For 0 < x < 4, f''(x) < 0, so the function is concave down on (0, 4). For x > 4, f''(x) > 0, so the function is concave up on (4, ∞).
So, the function f(x) = -x^6 + 6x^5 + 12x - 2 is concave up on (-∞, 0) and (4, ∞), and concave down on (0, 4).
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