Question

Let p(x) = x

2

(a ax), where a is constant and a > 0.

Find the local maxima and minima of p.

(Enter your maxima and minima as comma-separated

xvalue,classification pairs. For example, if you found that x =

=2 was a local minimum and x = 3 was a local maximum, you

should enter (-2,min), (3,max). If there were no maximum, you

must drop the parentheses and enter -2,min.)

maxima and minima:

What effect does increasing the value of a have on the x￾position of the maximum(s) you found? (Enter left, none or

right if it moves left, has no effect, or moves right.)

What effect does increasing the value of a have on the x￾position of the minimum(s) you found? (Enter left, none or

right if it moves left, has no effect, or moves right.)

What effect does increasing the value of a have on the y￾coordinate of the maximum(s) you found? (Enter up, none or

down if it moves up, has no effect, or moves down.)

Answer 1

 p(x)=x^2(a-x),

p'(x)=x(2a-3x),

x=0 and x=2a/3 

as a>0 so at x= 0 the second derivative is 2a > 0

so x=0 has a minimum, increasing the value of a will have no effect on the x-position of the minimum.

and at x= 2a/3 the second derivative = -2a <0

so at x= 2a/3 has a maximum value. Increasing the value of a will cause the x-position of the maximum to move to the right.

Increasing the value of a will cause the y-coordinate of the maximum to move down.