SOLUTION: If a = y^2 and b = y^4, and -1 ≤ y ≤ 3, then what is the largest possible difference between a and b?

Maximum values of continuous functions occur at values of the
independent variable where the derivative is 0 or at
endpoints of interval on which the variable is defined.

The difference is largest when  is the largest.

Let this difference be z.





Maximum values of continuous functions occur at values of the
independent variable (in this cases y) where the derivative is 0 or at
endpoints of interval on which the variable is defined.



We set that = 0







  








Using the 2nd derivative test:





Substituting y=0 gives 
which is positive so this gives a relative minimum value,
which is not the value we want

Substituting   gives 
which is negative so this value of y gives a relative maximum value.

To find this relative maximum value we substitute 
in the original equation 







This is a candidate for the correct answer, unless the endpoints
of the given interval give a larger value:

The end points of the given interval are -1 and 3

Substituting the endpoint -1




That's not larger than !

Substituting the endpoint 3




That's certainly not larger than !

Therefore the maximum difference between  and  is 

Edwin