Increasing And Decreasing Functions

Concept: A function is increasing for all if its derivative is always non-negative: Trigonometric expressions like are oscillatory and change sign unless restricted.

Step 1: Differentiate the function.

Step 2: Analyze the sign of . The expression: is a trigonometric function of the form: which always oscillates between positive and negative values unless identically zero. Thus, cannot remain non-negative for all .

Step 3: Conclusion. Since changes sign for all values of , the function cannot be increasing for all real .

Concept: A function is increasing if its derivative is non-negative for all in its domain: For a function of the form , derivative is: Since , the sign depends only on the numerator.
Step 1: {Differentiate numerator and denominator.}

Step 2: {Apply quotient rule.}

Step 3: {Simplify numerator.}
Expanding and simplifying:
Step 4: {Apply increasing condition.}
Since denominator is always positive,