Verified answer

Your function:

y = 1/(x² + 3)

To find the inflection points of the function, you have to find the second derivative of the function.

y = 1/(x² + 3)

First derivative (use chain rule or quotient rule):

y' = [(1)' * (x² + 3) - (x² + 3)' * (1)]/(x² + 3)²

y' = [(0) * (x² + 3) - (2x)(1)]/(x² + 3)²

y' = -2x/(x² + 3)²

Second derivative (again, chain rule or quotient rule):

y'' = [(-2x)' * (x² + 3)² - ((x² + 3)²)' * (-2x)]/[(x² + 3)²]²

y'' = [-2(x² + 3)² - (2(x² + 3) * (2x)) * (-2x)]/[(x² + 3)²]²

y'' = [-2(x² + 3)² + 8x²(x² + 3)]/[(x² + 3)²]²

y'' = [(x² + 3)(-2(x² + 3) + 8x²)]/[(x² + 3)²]² //Divide numerator and denominator by x² + 3.

y'' = [-2x² - 6 + 8x²]/[x² + 3]^3

y'' = [6x² - 6]/[x² + 3]^3

Solve for the zeros of the second derivative:

0 = [6x² - 6]/[x² + 3]^3

0 = 6x² - 6

0 = x² - 1

Your inflection points are at: x = 1, x = -1

You know that inflection points are points at which the graph changes concavity. An easy way to check the concavity of a function is by using a number line combined with a sign chart.

Test values that fall to the left and right sides of the inflection point by inputting values (that are not the values of the inflection points) into your function. This will tell you the concavity of those intervals.

Points to test: x = -2, 0, 2

y'' = [6x² - 6]/[x² + 3x]^3

y''(-2) = [6(-2)² - 6]/[(-2)² + 3]^3 -- This will give you a positive value for y''.

y''(0) = [6(0)² - 6]/[(0)² + 3]^3 -- This will give you a negative value for y''.

y''(2) = [6(2)² - 6]/[(2)² + 3]^3 -- This will give you a positive value for y''.

<──────(-1)──────(1)──────>

y'' positive───negative──positive

If the second derivative is positive, the function is concave up. If the second derivative is negative, the function is concave down.

The function is concave up at the intervals (-∞, -1) and (1, ∞). The function is concave down at the interval (-1, 1).

find the second derivative either use the chain rule or qoutient rule