Question

Find the inflection points of the function, if any.

Answer 1

\[f(x)=\frac{ 2x-8 }{ x-1 }\]

Answer 2

\[f'(x)=\frac{ 6 }{ (x-1)^2 }\]

Answer 3

\[f"(x)=-\frac{ 12 }{ (x-1)^3 }\]

Answer 4

I'm not sure what to do from here

Answer 5

You have concavity up when your second derivative is positive,
and have concavity down when your second derivative is negative,

Points of inflection occur where your second derivative is zero, ya?\[\large\rm 0=-\frac{12}{(x-1)^3}\]Inflection point is that special place between concave up and concave down.

Answer 6

Solve for x.

Answer 7

It looks like no solutions will exist, ya?

Answer 8

but when you graph the function there is both a concave up and down

Answer 9

you can type in the equation here: https://www.desmos.com/calculator

Answer 10

inflection points with f''(x) is zero or defined. ^^

Answer 11

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Answer 12

when the denominator is zero the function is undefined because you can't divide by zero. you have one possible inflection point. you do the second derivative test to make a sign chart to look at whether values to the left and right of the inflection point change signs. when the second derivative is positive, the function f(x) is concave up. when the second derivative is negative, the function f(x) is concave down. if the signs do not change, than that means the concavity didn't change.

Answer 13

Inflection points... x= ? some number.

Hope i helped!

Answer 14

could somebody show me how the chart would look?

Answer 15

I'm super confused

Answer 16

sure.

Answer 17

The second derivative test is to check and see where values change at the inflection point. You just pick a point smaller than your inflection point (like zero) and plug it into the second derivative to see if it is positive or negative. Than pick a value bigger than your inflection point (like 2) and plug it into the second derivative and see what the sign is. It is like the first derivative test.

Answer 18

So: The function is concave up (-infinity, 1) and concave down (1, infinity)

Answer 19

THANK YOU I UNDERSTAND IT NOW