Sketch the graph of the function. Indicate the transition points (local extrema & points of inflection).?

Question

y = x(1 - x)^1/3

(Please explain how the graph looks)

Answer 1

Since your ultimate goal is to find the local extrema and the points of inflection, find those first.

To do that you should use the first derivative of the function.

If y [or f(x)]= x(1 - x)^1/3, then:

dy/dx [or f '(x)]= (derivative of the first * the normal second) + (the normal first * the derivative of the second); therefore:

dy/dx = (1 * (1 - x)^1/3) + (x *(1/3(1 - x)^-2/3)(-1))

dy/dx = ((1 - x)^-2/3) * (((1 - x)^-1/2) - (1/3)x)

Now, set dy/dx = 0 and find all x values.

0 = ((1 - x)^-2/3) * (((1 - x)^-1/2) - (1/3)x)

dy/dx = 0 when x = .74999911; this is your critical point.

Now, you must use the first derivative line test.  Because dy/dx = 0 when x = .74999911, you know that this is a maximum or minimum (extrema).  To find out which one it is, pick particular points to the left and right of your critical point.  Try x = 0 and x = 1.

You find that by plugging 0 into dy/dx it equals approximately 1, and by plugging 1 into dy/dx, it equals -100.  Because the derivative is positive before the critical point and negative after the point, the first derivative line test proves that this critical point is the local maximum.  

When the derivative's values are positive, that means that the original function is increasing.  When the derivative's values are negative, that means that the function is decreasing.  

Plug the x-value of .74999911 into the original function, to find the cooresponding y-value.  

x = .74999911, y = .4725

This is your maximum and local extrema.

Now you have a general idea of what your graph looks like.  It is continually increasing until x = .74999911 where it reaches a height of .4725, then it begins to steadily decrease.

To find a point of inflection, you must find the second derivative.

If y [or f(x)] = x(1 - x)^(1/3) and

dy/dx [or (f '(x))]= [(1 - x)^-2/3) * (((1 - x)^-1/2) - (1/3)x], then

dy2/d2x [or f ''(x)] = [(1/3)(1 - x)^(-2/3)] + [(1 * ((-1/3)(1 - x)^-2/3)) + (x *(2/9)(1 - x)^(-5/3)(-1))]

= (-(2/3) * (1 - x)^(-2/3) - 2x/9 (1 - x)^(-5/3))

Set that equal to zero and it will give you your points of inflection.