Find the maximum and minimum values, if any, of the following functions given by
f(x) = |x + 2| - 1
Minimum value: -1 at x = -2. Maximum value: Does not exist.
g(x) = -|x + 1| + 3
Maximum value: 3 at x = -1. Minimum value: Does not exist.
h(x) = sin(2x) + 5
Maximum value: 6 (when sin(2x) = 1). Minimum value: 4 (when sin(2x) = -1).
f(x) = |sin 4x + 3|
Maximum value: 4 (when sin(4x) = 1). Minimum value: 2 (when sin(4x) = -1).
h(x) = x + 1, x ∈ (-1, 1)
Maximum value: Does not exist. Minimum value: Does not exist.
Explanation
This question tests understanding of absolute value functions, trigonometric functions, and the effect of domain restrictions on maxima/minima. Parts (i) and (ii) involve absolute value functions where the minimum of |x| is 0. Parts (iii) and (iv) use the property that -1 sin(θ) 1. Part (v) demonstrates that on an open interval, a strictly increasing function may not attain its supremum or infimum.