Find the inflection point, if any, of the function. (If an...

Find the inflection point, if any, of the function. (If an answer does not exist, enter DNE.) g(x) = 7x^3 - 6x (x, y) = ( 10. [-/1 Points] DETAILS 0/5 Submissions Used Find the inflection point, if it exists, of the function. (If an answer does not exist, enter DNE.) g(x) = 4x^3 - 6x^2 + 7x - 2 (x, y) = ( 11. [-/2 Points] DETAILS 0/5 Submissions Used Find the inflection point(s), if any, of the function. (If an answer does not exist, enter DNE.) g(x) = 2x^4 - 4x^3 + 4 smaller x-value (x, y) = ( larger x-value (x, y) = (

Transcript

00:01 We want to find the inflection point of g of x equals 7x cubed minus 6x.

00:05 So an inflection point is going to occur when a second derivative is equal to 0.

00:10 So first let's find the first derivative of g of x.

00:13 The derivative of the 7x cubed, that would be 7 times 3x squared, minus the derivative of 6x, that would just be 6, and this would simplify to 21x squared minus 6 for our first derivative.

00:30 Then let's do our second derivative, so that would be 21 times the derivative of x squared, which is 2x, and the derivative 6 is 0.

00:39 So that would just give us 42x for our second derivative.

00:46 And we just want to know when that is equal to 0.

00:50 So that's easy to solve.

00:52 We can divide both sides by 42, and we can see that x equals 0.

00:58 So we want to find the actual point, the x and the x and y coordinate so we know the x would be zero and we need to find g of zero that would be seven times zero cubed minus six times zero which we just give us zero so that would be the point zero zero can actually want to find the inflection point of g of x equals four x cubed minus six x squared plus seven x minus two so first we need to find the first derivative that would be four times the derivative of x cubed would be three x squared minus 6 times the derivative of x squared would be 2x plus the derivative of 7x we just be 7 and then the derivative of 2 is 0 and then if we simplify that that would give us 12 x squared minus 12 x plus 7 and if we do the second derivative that would be 12 times the derivative of x squared which is 2x minus the derivative of 12x would just be 12 we could simplify that to 25 x minus 12 and we want to know when is that second derivative equal to zero we could add 12 to both sides and get 24 x equals 12 divide both sides by 24 that gives us x equals 12 over 24 reduced to 1 half so that gives us our x coordinate is 1 half then to find our y coordinate we would need to find g of 1 half so we're just going to plug 1 half into that function so let's just get a calculator real quick so that would be four times one half cubed and then minus minus six times one half squared and then plus seven times one half and then minus two and that's going to give us our y value of 0 .5 which would be one half so that would be the point one half one half okay and lastly we want to find the inflection points of g of x equals 2 x the 4th minus 4x cubed plus 4...

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