Max, Min, & Point of Inflection?

Question

The question is: find the maximum, minimum and inflection points for each function.

2y = x^2 - 4x + 6

I am not used to having a number beside the "y". Should I divide it out then take the derivative? But how would I take the derivative of x^2/2 ?

Any help is greatly appreciated, thanks.

Answer 1

Yes, divide the two out first, you dont technically have to, but it will make your answers make more sense.



d(2*y)/dx = 2*dy/dx



the same thing holds for x^2/2



d(x^2/2)/dx = 1/2*d(x^2)/dx



To find the max and min, take the derivative and set it equal to zero. To determine whether it is a max or min you will need the second derivative. If it is positive at that point, it is a minimum, and if it is a negative at that point, it is a maximum. Then the inflection point is found by just setting the second derivative to zero.

Answer 2

The number before y does not affect the location of the max, min or points of inflection. But it does affect the y-value of the said points.



you can initially solve using x^2 -4x +6 simply to get the x-values. But return the said values to the original equation to get the y's.



Note: the derivative of k*f(x) = k*f '(x). Thus D(1/2*x^2) = (1/2)*2x = x.



Add: since your curve is a parabola it has a minimum, no maximum nor inflection points.

Answer 3

First you divide every term by 2 and get:



y = x^2/2 -2x + 3



To get the derivative:



y´ = 2x/2 - 2 = x -2 ... the root of y´ is x=2



Studying the signal of y´you get: negative before 2

positive after 2



Then there is a maximum in x = 2 and y = 1



Getting the second derivative: y´´ = 1 constant... then there is no root ... and for that no inflection point.

Answer 4

I am not used to having a number beside the "y". Should I divide it out then take the derivative? => that is a correct way to do it.



But how would I take the derivative of x^2/2 ?

can you take the derivative of x^2 ? the coefficient in this case is 1. In your case, the coef. is 1/2, so just use the same rule when you take derivative or x^2.



key: power rule.

Answer 5

Divide both sides of the equation by 2, which gives:



y=1/2 x^2-2x+3



dy/dx=x-2

Answer 6

first divide the equation by 2

dy/dx=x-2

then equate dy/dx to zero

x=2

take the second derivative

d^2y/dx^2=1

that is positive

so that is the minimum point of the curve

and the cooridinate of minimum point is (2,3)

Answer 7

i do no longer think of it rather is achievable... factors of inflection happen the place the concavity of the graph differences instructions, yet at a max the graph is often concaved down, and at a min the graph is often concaved up.

Answer 8

yes you are right divide everythign by 2. and d/dx (x^2/2)=x