Problem: For values of x less than 1 but greater than −4 , the expression 2x−2x2−2x+2 has:
Answer Choices:
A. no maximum or minimum value
B. a minimum value of +1
C. a maximum value of +1
D. a minimum value of -1
E. a maximum value of -1
Solution:
y=21x−1x2−2x+2=21[x−1+x−11]
The sum of a number and its reciprocal is numerically least when the number is ±1.
For x−1=1, we get x=2, which is excluded.
∴x−1=−1 and y=−1
All other values of x in the given interval yield values of y less than −1.
dxdy=(2x−2)2(2x−2)(2x−2)−(x2−2x+2)(2)=0
x=0,x=2 (excluded)
At x=21, we find dxdy is negative.
At x=−21, we find dxdy is positive.
Therefore, y is a maximum when x=0.