The range of parameter b, for which the function f(x)=∫0^x(b l^2+b+cost) d t is entirely increas

step 1 In this question we have to find range of parameter B for which function is completely increasin

The range of parameter b, for which the function f(x)=∫0^x(b...

The range of parameter $b$, for which the function
$f(x)=\int_{0}^{x}\left(b l^{2}+b+\cos t\right) d t$
is entirely increasing or decreasing for all real values of $x$ is
(A) $[-1,1]$
(B) $(-\infty,-1] \cup[1, \infty)$
(C) $(-\infty,-1) \cup(1, \infty)$
(D) $(-1,1)$

Key Concepts

Parameter Constraints in Inequalities

When a derivative contains both a parameter-dependent polynomial and a bounded trigonometric component, finding the range of the parameter involves setting up and solving inequalities that ensure the derivative remains always nonnegative or nonpositive. This process demonstrates the interplay between algebraic and trigonometric considerations in controlling the behavior of a function.

Bounded Nature of the Cosine Function

The cosine function, being a trigonometric function, has the important property of being bounded between -1 and 1. This boundedness is key when it is combined with other expressions, as it allows us to establish inequalities which determine when the overall expression maintains a constant sign.

Monotonicity via the Sign of the Derivative

A function is increasing if its derivative is nonnegative everywhere and decreasing if its derivative is nonpositive everywhere. By ensuring that the derivative does not change sign over the entire real line, one guarantees that the function is monotonic.

Fundamental Theorem of Calculus

This theorem connects differentiation and integration by stating that if a function is defined as an integral with a variable upper limit, then its derivative is the integrand evaluated at that upper limit. This relationship helps determine properties of the original function based solely on its integrand.

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