Let a+b=4, a<2 and g(x) be a monotonically increasing...

Let $a+b=4, a<2$ and $g(x)$ be a monotonically increasing function of $x$. Then, $f(a)=\int_{0}^{a} g(x) d x+\int_{0}^{b} g(x) d x$
(A) increases with increase in $(b-a)$
(B) decreases with increase in $(b-a)$

Key Concepts

Differentiation of Integral Functions

Differentiation of functions defined by integrals, particularly with variable upper limits, involves applying the Fundamental Theorem of Calculus in conjunction with the chain rule. This process is used to analyze how small changes in the limits affect the overall function, and it is a key tool in assessing whether the function increases or decreases as its input changes.

Using Constraints to Relate Variables

Constraints, such as a constant sum between two variables, create a relationship between them that can be exploited to simplify and analyze complex functions. By expressing one variable in terms of another through such a constraint, the behavior of functions defined with multiple integrals can be studied more effectively and reduced to a single variable dependence.

Definite Integrals

Definite integrals provide a way to accumulate the values of a function over a specified interval. They represent the net area under the curve of the function between the limits of integration. This concept is central to understanding how changing the limits of integration affects the resulting value of the integral.

Monotonic Functions

A monotonic function, specifically one that is monotonically increasing, is a function where an increase in the input always produces an increase (or at least not a decrease) in the output. This property guarantees that the function preserves the order of its inputs, which is critical when comparing values or analyzing the behavior of integrals that depend on such a function.

Variable Limits of Integration

When the limits of integration depend on a variable, the value of the definite integral becomes a function of that variable. This means that any variation in the limit, subject to certain conditions or constraints, establishes a relationship between the variable and the accumulated area, resulting in a function that can be analyzed for increasing or decreasing behavior.

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