If f^'(x)>0 and g^'(x) fog(x+1) (B) fog(x)>fog(x-1) (C) gof(x)>gof(x+1)

step 1 So in the question f -diss x is greater than 0 and g -dice x is less than 0 this implies fx is i

If f^'(x)>0 and g^'(x)<0 ∀x ∈R, then (A) ∫og(x)>fog(x+1) (B)...

If $f^{\prime}(x)>0$ and $g^{\prime}(x)<0 \forall x \in R$, then
(A) $\int \operatorname{og}(x)>\operatorname{fog}(x+1)$
(B) $\operatorname{fog}(x)>\operatorname{fog}(x-1)$
(C) $\operatorname{gof}(x)>\operatorname{gof}(x+1)$
(D) $g o f(x)>g o f(x-1)$

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Key Concepts

Function Composition and Its Monotonicity

Function composition involves applying one function to the results of another (denoted as f ? g for f(g(x))). The monotonicity of a composite function depends on the individual monotonicities of the functions involved. For example, if one function is increasing and the other is decreasing, the composite may reverse the usual order of inequalities. Understanding how these properties combine is crucial when comparing the outputs of composite functions at shifted inputs.

Monotonic Functions

A monotonic function is one that either never decreases or never increases as its input increases. In this context, a strictly increasing function (where the derivative is positive everywhere) ensures that larger input values always lead to larger outputs, while a strictly decreasing function (with a negative derivative everywhere) means larger input values lead to smaller outputs. This is a foundational concept for comparing function values at different points.

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