f(x)= x^3/x^2-4
defined on the interval [-18,20]
f(x) is concave up on the region ? to ? and ? to ?
the inflection points are ?, ? and ?
I assume you mean f(x) = x^3/(x^2-4)
First find the points where f'(x) = 0
f'(x) = [(x^2-4)(3x^2)-2x^4]/(x^2 -4)^2
(x^2-4)(3x^2)-2x^4 = 0 when f'(x) = 0
3x^4 -12 x^2 -2x^4 = 0
x^2 (x^2-12)
x = 0, +2sqrt3 and -2sqrt3
Now evaluate the second derivative at those three critical points. The function is concave up if f''(x) > 0 and is an inflection point if f''(x) = 0
To determine where the function f(x) = x^3/(x^2-4) is concave up and find its inflection points on the interval [-18,20], we will follow these steps:
Step 1: Find the second derivative of f(x).
Step 2: Set the second derivative equal to zero to find the potential inflection points.
Step 3: Analyze the sign of the second derivative to determine the concavity.
Let's begin with step 1:
Step 1: Find the second derivative of f(x).
To find the second derivative, we will apply the quotient rule. The quotient rule states that if we have a function f(x) = u(x)/v(x), where u(x) and v(x) are differentiable functions, then the second derivative is given by:
f''(x) = (u''(x)v(x) - u(x)v''(x)) / (v(x))^2
For f(x) = x^3/(x^2-4), we can rewrite the function as:
f(x) = x^3 * (x^2-4)^-1
Applying the quotient rule, we differentiate the numerator and denominator:
u(x) = x^3
u''(x) = 6x
v(x) = (x^2-4)
v''(x) = 2x
Then, substituting these values into the quotient rule formula, we get:
f''(x) = (6x * (x^2-4) - x^3 * 2x) / (x^2-4)^2
Simplifying further:
f''(x) = (6x^3 - 24x - 2x^4) / (x^2-4)^2
This is the second derivative of f(x).
Step 2: Set the second derivative equal to zero to find the potential inflection points.
To find the potential inflection points, we set the second derivative equal to zero:
f''(x) = 0
(6x^3 - 24x - 2x^4) / (x^2-4)^2 = 0
We solve this equation to find the values of x that make the second derivative equal to zero. These values will be the potential inflection points.
Step 3: Analyze the sign of the second derivative to determine the concavity.
To analyze the concavity of the function, we need to determine the sign of the second derivative in different intervals.
To do this, we will consider three intervals: (-∞, -2), (-2, 2), and (2,∞) which are derived from the factors in the original function.
For each interval, we pick a test point within that interval and plug it into the second derivative. Based on the sign of the result, we can determine the concavity.
For example, let's choose x = -3, 0, and 3 as test points (one from each interval):
When x = -3:
f''(-3) = (6(-3)^3 - 24(-3) - 2(-3)^4) / ((-3)^2-4)^2 = -72 / 49 < 0
When x = 0:
f''(0) = (6(0)^3 - 24(0) - 2(0)^4) / (0^2-4)^2 = 0 / 16 = 0
When x = 3:
f''(3) = (6(3)^3 - 24(3) - 2(3)^4) / ((3)^2-4)^2 = 72 / 49 > 0
Based on these calculations, we can conclude the following:
f(x) is concave up on the intervals (-∞, -2) and (2, ∞).
To determine the exact locations of the inflection points, we need to solve the equation f''(x) = 0 that we obtained earlier:
(6x^3 - 24x - 2x^4) / (x^2-4)^2 = 0
Simplifying the equation further may help identify possible solutions:
6x^3 - 24x - 2x^4 = 0
Since this is a cubic equation, finding exact solutions might be challenging. Depending on the level of precision needed, you may have to use numerical methods, such as graphing the equation or using software/tools to find the approximations of the inflection points.
With this approach, we have determined where f(x) is concave up and found the equation to identify the inflection points.