Question

If x ∈ [0, 1], and f(x) and g(x) are defined as: f(x) = sin(cos(πx/4)) and g(x) = cos(sin(πx/4)) then:

• A. f is monotonic increasing and g is monotonic decreasing.
• B. f is monotonic increasing and g is monotonic increasing.
• C. f is monotonic decreasing and g is monotonic decreasing.
• D. f is monotonic decreasing and g is monotonic increasing.

Hint:

Analyze monotonicity by studying the behavior of nested trigonometric functions over the given interval.

Answer 1

The Correct Option is D

For f (x) = sin(cos(πx/4)), an increase in x within [0, 1] causes cos(πx/4) to decrease. As sin is an increasing function, f (x) consequently decreases. For g(x) = cos(sin(πx/4)), an increase in x leads to sin(πx/4) increasing. Given that cos is a decreasing function, g(x) increases.