what are the inflection points of 5x^2-45/x^2-25
what is the numerator and what is the denominator?
assuming the usual sloppiness, we have
y=(5x^2-45)/(x^2-25)
y'= -160x/(x^2-25)^2
y" = 160(3x^2+25)/(x^2-25)^3
since y" is never zero, there are no inflection points, as can be confirmed here:
http://www.wolframalpha.com/input/?i=%285x^2-45%29%2F%28x^2-25%29
To find the inflection points of the function f(x) = (5x^2-45)/(x^2-25), we need to follow these steps:
1. Find the second derivative of f(x).
2. Set the second derivative equal to zero and solve for x.
3. Identify the x-values obtained in step 2 as potential inflection points.
4. Test the concavity of f(x) around these x-values to confirm if they are inflection points.
Let's go through each step in detail:
1. Find the second derivative of f(x):
To find the second derivative, we need to differentiate the function twice. The first derivative of f(x) is found using the quotient rule:
f'(x) = [ (2x)(x^2-25) - (5x^2-45)(2x) ] / (x^2-25)^2
Now, simplify the equation to determine f''(x), the second derivative of f(x).
2. Set the second derivative equal to zero and solve for x:
Set f''(x) = 0, and solve for x. The obtained x-values will be the potential inflection points.
3. Identify the x-values obtained as potential inflection points:
The x-values obtained in step 2 will be the potential inflection points. Note that these points are only possible inflection points and need to be tested further for confirmation.
4. Test the concavity of f(x) around the potential inflection points:
To confirm if the potential inflection points are indeed inflection points, we need to examine the concavity of f(x) around these points. To do this, we can take a value of x that lies between two potential inflection points and substitute it into the second derivative equation.
- If the second derivative is positive at a specific x-value, the function is concave up at that point.
- If the second derivative is negative at a specific x-value, the function is concave down at that point.
An inflection point occurs when the concavity changes, meaning the second derivative changes sign.
By following these steps, you can determine the inflection points of the given function.