Question

Find the inflection points for F(x)=1/4X^4-x^3+6

Answer 1

Inflection points are where the function takes on the greatest acceleration; that means that the function is changing the fastest or steepest at that point. In order to figure this out, we need to take the 2nd derivative of the function, set it equal to 0, and solve for (x).

F(x) = 1/4 (x^4) - x^3 + 6

by the Power Rule:

F'(x) = x^3 - 3x^2

Lets derive again, and by the Power Rule again:

F''(x) = 3x^2 - 6x

Now we need to set the equation to 0:

0 = 3x^2 - 6x

0 = 3(x^2 - 6)

0 = x^2 - 6
\[0 = (x+\sqrt{6})(x-\sqrt{6})\]
There are two (x) that pop out, so there are two inflection points on the function: \[x = \pm \sqrt{6}\]

Answer 2

should be square root of 3, made a factoring mistake

Answer 3

The second derivative of F(x) is:\[3 x^2-6 x \] When set to zero and solved for x,\[\{x=0,x=2\} \]

Answer 4

A plot is attached.