Find the inflection points of f(x)=8x^4+46x^3-18x^2+1. (Give your answers as a comma separated list, e.g., 3,-2.)

Please Help!!!!!

what's the problem? Inflection points occur where f'' = 0

f' = 32x^3 + 138x^2 - 36x
f'' = 96x^2 + 276x - 36
= 8(12x^2 + 23x - 3)
f''=0 where x = -3, 1/8

Find the points of inflection of the graph of the function. (If an answer does not exist, enter DNE.)

f(x) = sin
x
4
, [0, 8π]

To find the inflection points of a function, we need to find the points where the concavity of the function changes. In mathematical terms, we are looking for points where the second derivative of the function changes sign.

Let's start by finding the first and second derivatives of the given function.

Given function: f(x) = 8x^4 + 46x^3 - 18x^2 + 1

First derivative (f'(x)):
f'(x) = 32x^3 + 138x^2 - 36x

Second derivative (f''(x)):
f''(x) = 96x^2 + 276x - 36

To find the inflection points, we need to solve the equation f''(x) = 0.

Setting f''(x) = 0:
96x^2 + 276x - 36 = 0

We can solve this quadratic equation using the quadratic formula.

x = (-b ± √(b^2 - 4ac)) / (2a)

In this case:
a = 96
b = 276
c = -36

Solving for x gives us two potential values. Let's calculate them:

x = (-276 ± √(276^2 - 4 * 96 * -36)) / (2 * 96)

Simplifying this equation will give us the values of x, which are potential inflection points.

Calculating the values:
x ≈ -2.33
x ≈ 0.16

So, the inflection points of the given function are approximately -2.33 and 0.16.

Therefore, the inflection points of f(x) = 8x^4 + 46x^3 - 18x^2 + 1 are approximately -2.33 and 0.16.