Find the interval of concavity for f(x)=e^x/(5+e^x)?
f'=[5*e^x ]/ [5+e^x]^2 & f"=[5*e^x ][5-e^x]/ [5+e^x]^3
To find the interval of concavity for the function f(x) = e^x/(5+e^x), we need to analyze the second derivative, f"(x), and determine where it is positive or negative.
Given that f"(x) = [5*e^x ][5-e^x]/ [5+e^x]^3, let's simplify it further.
Since e^x is always positive, we can remove it from the equation and focus on the remaining terms:
f"(x) = [5][5-e^x]/ [5+e^x]^3
Now, let's analyze the sign of the numerator and denominator separately.
Numerator: [5][5-e^x]
Since 5 is positive, the sign of the numerator depends on (5 - e^x). If e^x > 5, then the numerator is negative. If e^x < 5, then the numerator is positive.
Denominator: [5+e^x]^3
Since both 5 and e^x are positive, the denominator is always positive.
Now, we can determine the sign of f"(x) based on the combination of the numerator and denominator.
If (5 - e^x) < 0 and [5+e^x]^3 > 0, then f"(x) < 0.
If (5 - e^x) > 0 and [5+e^x]^3 > 0, then f"(x) > 0.
To summarize, the intervals of concavity for the function f(x) = e^x/(5+e^x) are:
- The function is concave down (f"(x) < 0) when e^x > 5.
- The function is concave up (f"(x) > 0) when e^x < 5.
So, the interval of concavity is: x < ln(5) for concave up, and x > ln(5) for concave down.