Let f(t)=|t-1|-|t|+|t+1|, ∀t ∈R and g(x)=max{f(t): x+1 ≤t...

Let $f(t)=|t-1|-|t|+|t+1|, \forall t \in R$ and
$g(x)=\max \{f(t): x+1 \leq t \leq x+2\} ; \forall x \in R$. Find $g(x)$
and the area bounded by the curve $y=g(x)$, the $X$ -axis and the lines $x=-3 / 2$ and $x=5$.

Key Concepts

Definite Integration for Area Calculation

Definite integration is used to compute the area under a curve between two given bounds. This technique sums up infinitely many small slices of area to determine the total area bounded by the function and the axis. It is an essential concept in calculus with numerous applications, including finding regions defined by curves and other boundaries.

Optimization on a Closed Interval

Optimization techniques on closed intervals entail finding the maximum or minimum values of a function within a bounded range. With continuous functions, the Extreme Value Theorem guarantees that such maxima and minima exist. The process typically involves evaluating the function at critical points—where the derivative equals zero or does not exist—as well as at the endpoints of the interval.

Function Transformation

Function transformations, such as shifting or stretching a function, or considering derived functions like one defined as the maximum value over an interval, are critical in mathematical analysis. Transformations allow us to understand how changes to input variables affect the output and how properties such as symmetry, periodicity, or extremal values are preserved or modified.

Absolute Value Functions

Absolute value functions return the non-negative magnitude of a real number, which causes expressions involving absolute values to be naturally split into different cases based on the sign of the internal expressions. This is a fundamental tool in analyzing and simplifying functions that might otherwise have discontinuous definitions or complex behavior within different intervals.

Piecewise Function Analysis

Many functions defined using absolute value expressions can be rewritten as piecewise functions. This process involves determining the intervals on which each expression inside an absolute value is positive or negative, thereby allowing the function to be expressed in simpler, often linear, pieces. This method is crucial when analyzing the overall behavior of a function, solving equations, or optimizing the function over an interval.

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