Answer the following questions for the function
f(x) = \frac{ x^3 }{ x^2 - 36 }
defined on the interval [ -20, 16 ].
A. The function f(x) has vertical asympototes at____ and___
B. f(x) is concave up on the region___ to and____ to___
C. The inflection point for this function is when x = ____
f(x)=(x^3)/(x^2-36)
To answer these questions for the given function, we need to analyze the properties of the function f(x) = x^3 / (x^2 - 36) on the interval [-20, 16].
A. Vertical asymptotes:
Vertical asymptotes occur when the denominator of a fraction becomes zero. In this case, the denominator (x^2 - 36) becomes zero when x = -6 and x = 6. These are the two vertical asymptotes for the function.
B. Concavity:
To find where the function is concave up or concave down, we need to consider the second derivative of the function. Let's find the first and second derivatives first:
First derivative:
f'(x) = (3x^2(x^2 - 36) - x^3(2x)) / (x^2 - 36)^2
Simplifying further, we get:
f'(x) = (3x^4 - 108x^2 - 2x^4) / (x^2 - 36)^2
f'(x) = (x^4 - 108x^2) / (x^2 - 36)^2
Second derivative:
f''(x) = [(x^4 - 108x^2)(2x^2-72) - (4x^3)(x^2-36)] / (x^2 - 36)^4
Simplifying further, we get:
f''(x) = (2x^6 - 72x^4 - 216x^2) / (x^2 - 36)^3
Now, we need to determine when the second derivative is positive or negative.
For f''(x) > 0, we have: (2x^6 - 72x^4 - 216x^2) / (x^2 - 36)^3 > 0
Simplifying further, we obtain: (x^2 - 12)(2x^4 + 24x^2 + 18) > 0
Analyzing the sign changes in the expression, we find that f''(x) > 0 when x < -6 and -sqrt(3) < x < sqrt(3).
Therefore, f(x) is concave up on the region [-20, -6] and on the region (-sqrt(3), sqrt(3)], as required.
C. Inflection point:
An inflection point occurs when the concavity of a function changes. We have already determined that f(x) is concave up on the region (-sqrt(3), sqrt(3)).
Now, to find the inflection point, we need to check where the second derivative changes sign within that region. We can observe that f''(x) changes sign at x = 0, indicating that there is an inflection point at x = 0.
Therefore, the inflection point for this function is when x = 0.