Analyzing Piecewise Functions for Increasing and Decreasing Behavior

The question asks us to evaluate two statements regarding the increasing and decreasing nature of a given piecewise function \({\rm{f}}\left( {\rm{x}} \right)\) over specific intervals.

The function is defined as:

\[{\rm{f}}\left( {\rm{x}} \right) = \left\{ {\begin{array}{*{20}{c}} {3{{\rm{x}}^2} + 12{\rm{x}} - 1,{\rm{\;\;}} - 1 \le {\rm{x}} \le 2}\\ {37 - {\rm{x}},{\rm{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}}2 < {\rm{x}} \le 3} \end{array}} \right.\]

Statement 1: f(x) is increasing in the interval [-1, 2]

To determine if a function is increasing in an interval, we examine its derivative in that interval. For the interval \([-1, 2]\), the function is \({\rm{f}}\left( {\rm{x}} \right) = 3{{\rm{x}}^2} + 12{\rm{x}} - 1\). We need to find the derivative \({\rm{f}}'\left( {\rm{x}} \right)\).

The derivative is:

\[{\rm{f}}'\left( {\rm{x}} \right) = \frac{d}{dx}\left( {3{{\rm{x}}^2} + 12{\rm{x}} - 1} \right)\] \[{\rm{f}}'\left( {\rm{x}} \right) = 6{\rm{x}} + 12\]

For the function to be increasing in the interval \([-1, 2]\), we need \({\rm{f}}'\left( {\rm{x}} \right) \ge 0\) for all \({\rm{x}}\) in \([-1, 2]\). Let's check this condition:

\[6{\rm{x}} + 12 \ge 0\] \[6{\rm{x}} \ge -12\] \[{\rm{x}} \ge \frac{-12}{6}\] \[{\rm{x}} \ge -2\]

The condition for \({\rm{f}}\left( {\rm{x}} \right)\) to be increasing is \({\rm{x}} \ge -2\). The interval given in the statement is \([-1, 2]\). All values of \({\rm{x}}\) in the interval \([-1, 2]\) satisfy the condition \({\rm{x}} \ge -2\). Therefore, the function \({\rm{f}}\left( {\rm{x}} \right)\) is indeed increasing in the interval \([-1, 2]\).

Statement 1 is correct.

Statement 2: f(x) is decreasing in the interval (2, 3]

To determine if a function is decreasing in an interval, we examine its derivative in that interval. For the interval \((2, 3]\), the function is \({\rm{f}}\left( {\rm{x}} \right) = 37 - {\rm{x}}\). We need to find the derivative \({\rm{f}}'\left( {\rm{x}} \right)\).

The derivative is:

\[{\rm{f}}'\left( {\rm{x}} \right) = \frac{d}{dx}\left( {37 - {\rm{x}}} \right)\] \[{\rm{f}}'\left( {\rm{x}} \right) = -1\]

For the function to be decreasing in the interval \((2, 3]\), we need \({\rm{f}}'\left( {\rm{x}} \right) \le 0\) for all \({\rm{x}}\) in \((2, 3]\). The derivative \({\rm{f}}'\left( {\rm{x}} \right) = -1\), which is always less than or equal to 0 for all values of \({\rm{x}}\). Therefore, the function \({\rm{f}}\left( {\rm{x}} \right)\) is decreasing in the interval \((2, 3]\).

Statement 2 is correct.

Conclusion on Statements

Based on our analysis:

• Statement 1 is correct because \({\rm{f}}'\left( {\rm{x}} \right) = 6{\rm{x}} + 12 \ge 0\) for all \({\rm{x}} \in [-1, 2]\).
• Statement 2 is correct because \({\rm{f}}'\left( {\rm{x}} \right) = -1 \le 0\) for all \({\rm{x}} \in (2, 3]\).

Since both statements are correct, the appropriate option is the one indicating that both 1 and 2 are correct.

Revision Table: Function Monotonicity

Additional Information: Piecewise Functions and Derivatives

A piecewise function is defined by multiple sub-functions, each applying to a specific interval of the domain. To analyze the behavior of a piecewise function, like finding where it is increasing or decreasing, you must analyze each piece separately over its defined interval.

The derivative helps determine the slope of the tangent line at any point on the function's curve. A positive derivative indicates an increasing function, while a negative derivative indicates a decreasing function. At points where the derivative is zero, the function may have a local maximum, minimum, or a point of inflection (though this needs further checks).

For a piecewise function, you calculate the derivative for each piece. Special attention might be needed at the points where the definition of the function changes (the boundary points between intervals) to check for differentiability and continuity, although this question only asks about intervals where the function is smooth within each piece.