f(x)= x/(1+x)^2
c) find the intervals of concavity and the inflection points.
My answers:
f is concave downward on (- infinity, 2) and concave upward on (2, infinity).
Inflection point: ( 2, 2/9)
f(x)=x*(1+x)^-2
f'= (1+x)^-2 -2x(1+x)^-3
f"=-2(1+x)^-3-2(1+x)^-3+6x(1+x)^-4
so where is f"=0? Setting it to zero, multiplying through by (1+x)^4 then
0=-2(1+x)-2(1+x)+6x
0=-2x-2x+6x -4
x= 2
for the inflection point, I didn't do y.
now for x<2, f"=negative, so concave downward, for x>2 upward.
To find the intervals of concavity and the inflection points of the function f(x) = x/(1+x)^2, we'll need to follow these steps:
1. Find the second derivative of the function.
2. Set the second derivative equal to zero and solve for x to find any possible inflection points.
3. Analyze the signs of the second derivative to determine the intervals of concavity.
4. Substitute the x-values found from step 2 into the original function to find the corresponding y-values for the inflection points.
Let's go through each of these steps:
1. Finding the second derivative:
The first derivative of f(x) is given by: f'(x) = [(1+x)^2 - 2x(1+x)] / (1+x)^4
To find the second derivative, differentiate f'(x) with respect to x:
f''(x) = [((1+x)^2 - 2x(1+x))/(1+x)^4]' = [((1+x)^2 - 2x(1+x))' / (1+x)^4]
Simplifying, we get:
f''(x) = [(2(1+x) - 2(1+x) - 2x) / (1+x)^4] = -2 / (1+x)^3
2. Finding the possible inflection points:
Set f''(x) equal to zero and solve for x:
-2 / (1+x)^3 = 0
Since a fraction is equal to zero only when the numerator is zero, we have:
-2 = 0
However, this equation has no solution, which means there are no inflection points for the given function.
3. Analyzing the signs of the second derivative:
The signs of the second derivative will determine the intervals of concavity.
For f''(x) = -2 / (1+x)^3:
- If (1+x)^3 < 0, then f''(x) > 0, and the function is concave upward.
- If (1+x)^3 > 0, then f''(x) < 0, and the function is concave downward.
Since the denominator of f''(x) is always positive, we only need to consider the sign of -2 to determine concavity.
For -2:
- If -2 < 0, then f''(x) > 0, and the function is concave upward.
- If -2 > 0, then f''(x) < 0, and the function is concave downward.
From this, we can conclude that f is concave upward on the interval (-∞, 2) and concave downward on the interval (2, ∞).
4. Finding the corresponding y-values for the inflection points:
Since there are no inflection points for this function, we do not need to find any corresponding y-values.
Therefore, your answers are partially correct:
f is concave downward on (-∞, 2) and concave upward on (2, ∞).
However, there is no inflection point for this function.
I hope this explanation helps! Let me know if you have any further questions.