What is the least possible value of (--2x+1)^2 if -2 is less than or equal to x less than or equal to 2?
Let y = (-2x + 1)²
and we can immediately see that y cannot be negative since its evaluation always involves squaring. It therefore follows that the least possible value of y in general would be 0.
But if we now consider the interval -2 ≤ x ≤ 2, we need to check if the value of x for which y = 0 falls into this interval.
Now, when y = 0
(-2x + 1)² = 0
-2x + 1 = 0
-2x = -1
x = 1/2
which clearly falls inside the interval, and so the least possible value of (-2x + 1)² is 0.
If the required interval was, say, 1 ≤ x ≤ 2 then we cant accept this value so we have to investigate and find that the function is always increasing in the interval and so the least value would be the value at x = 1 which would be 1.