Consider the following function

Question

Consider the function f(x) = x^2/(8(x^2)+3)

List the x values of the inflection points of f. If there are no inflection points, enter 'NONE'.

Answer 1

It is important to understand functions as much as get answers to questions.

Note that this function is symmetric, has a minimum at (0,0), and has a horizontal asymptote of (1/8), increasing for positive x and decreasing for negative x, so the function is convex at (0,0) and concave in the tails, meaning two inflection points symmetric about 0.

Now, to find them:

f(x) = x^2/(8x^2 + 3) = 1/8 - 3/(8(8x^2+3))

f'(x) = 6x/(8x^2+3)²

f''(x) = (6(8x^2+3)-192x^2)/(8x^2+3)³ = (18 - 144x^2)/(8x^2+3)³ = 18(1 - 8x^2)/(8x^2+3)³

As the denominator is always positive, the sign of f''(x) is the sign of the numerator, a concave parabola with 0s at ±√(1/8) or, if you prefer, 0s at ±√2 / 4

Thus, the inflection points are at x = ±√2 / 4; y = x^2/(8x^2+3) = 1/8(8*1/8+3) = 1/8/4 = 1/32

Inflection points are (-√2 / 4, 1/32) and (√2 / 4, 1/32)