how can we make a graph of relative maxima and relative minima??

Question

i am really very confused...while making a graph of that function which may contain any power...for example...x^3 and 3x^4-16x^3+18x^2

Answer 1

Find relative extremes by taking the derivative of the function and setting it equal to zero.

Your examples were
x^3
derivative: 3x^2
zeros: x=0
obviously you have to check each zero, in this case it is not really a local extreme of the function, because the 2nd derivative, 6x, is zero when x=0.

3x^4 - 16x^3 + 18x^2
derivative: 12x^3 - 48x^2 + 36x
factor: 12x(x - 1)(x - 3)
zeros: x=0, x=1, x=3
Test them using the 2nd derivative, 36x^2 - 96x + 36
36*0 - 96*0 + 36 = 36
positive, x=0 is a local minimum
36*1 - 96*1 + 36 = -24
negative, x=1 is a local maximum
36*9 - 96*3 + 36 = 72
positive, x=3 is a local minimum

Basically... each zero of the first derivative is a possible local extreme of the function. Put each one into the second derivative... if the second derivative is positive, it is a local min; negative, a local max. If the second derivative is zero, it is not a local extreme.

If you are wanting to graph them, just make sure your x-axis contains all of the local values and then find out what sort of y-values you are looking at to make sure the curve will fit.

Answer 2

Look for places where the 1'st derivative of the function is zero. These are the relative maxima and minima. If the 2'dn derivative at these points is negative, it's a maxima. If the 2'nd derivative is positive, it's a minima.

As for making a graph at that point, it's just like any other graph. y = F(x) where F is (in your case) some polynomial function. Then choose values of x in the vicinity of the maxima or minima you wish to graph, compute F(x) for those x values, plot the resultant (x, F(x)) points, and connect them with a smooth curve.


Doug

Answer 3

ok, Gerry gave you the zeros. Now we could comprehend while it has a optimal or minimal cost. to discover this out, we turn to the spinoff: f(x) = x^3 + 4x^2 - 5x f'(x) = 3x^2 + 8x - 5 x = (-8 +/- sqrt(sixty 4 + 60)) / 6 x = (-8 +/- sqrt(124)) / 6 x = (-8 +/- 2 * sqrt(31)) / 6 x = (-4 +/- sqrt(31)) / 3