find the intervals on which the curve y=x/(x+1)^2 is concave upward or concave downward.
To determine the intervals on which the curve is concave upward or concave downward, we need to find the second derivative of the curve and analyze its sign.
Step 1: Find the first derivative.
Differentiate the function y = x/(x+1)^2 using the quotient rule. The first derivative is given by:
dy/dx = [(x+1)^2 * (1) - (x) * (2(x+1))]/(x+1)^4
= [(x+1) - 2x(x+1)]/(x+1)^4
= (x+1 - 2x^2 - 2x)/(x+1)^4
= (-2x^2 - x + 1)/(x+1)^4
Step 2: Find the second derivative.
To find the second derivative, differentiate the first derivative obtained in step 1.
d^2y/dx^2 = [(-2x^2 - x + 1)'(x+1)^4 - (-2x^2 - x + 1)(x+1)^4']/(x+1)^8
Simplifying this expression, we get:
d^2y/dx^2 = [(-4x - 1)(x+1)^4 - (-2x^2 - x + 1)(4(x+1)^3)]/(x+1)^8
Step 3: Analyze the sign of the second derivative.
To determine the intervals of concavity, we need to know where the second derivative is positive (concave upward) and where it is negative (concave downward). To do this, we can examine the intervals where the second derivative is greater than zero or less than zero.
Let's solve the inequality:
d^2y/dx^2 > 0 (for concave upward)
(-4x - 1)(x+1)^4 - (-2x^2 - x + 1)(4(x+1)^3) > 0
d^2y/dx^2 < 0 (for concave downward)
(-4x - 1)(x+1)^4 - (-2x^2 - x + 1)(4(x+1)^3) < 0
By solving these inequalities, we can find the intervals on which the curve is concave upward or concave downward.