Given that f^'(x)>g^'(x) for all real x and f(0)=g(0), then (A) f(x)>g(x) ∀x ∈(0, ∞) (B) f(x)<g(

step 1 So it is given that F -dach x is greater than G -dash -X for all -dial x. So let let H -X equal

Given that f^'(x)>g^'(x) for all real x and f(0)=g(0), then...

Given that $f^{\prime}(x)>g^{\prime}(x)$ for all real $x$ and $f(0)=g(0)$, then
(A) $f(x)>g(x) \forall x \in(0, \infty)$
(B) $f(x)<g(x) \forall x \in(-\infty, 0)$
(C) $f(x)<g(x) \forall x \in(0, \infty)$
(D) $f(x)>g(x) \forall x \in(-\infty, 0)$

Key Concepts

Monotonicity

Monotonicity refers to the property of a function being exclusively non-decreasing or non-increasing over an interval. When the derivative of a function is consistently positive (or, in comparison, the difference of derivatives is positive), the function exhibits strict monotonic behavior, ensuring that it increases or decreases consistently.

Derivative Comparison

This concept involves comparing the rates of change of two functions. When one function's derivative is always greater than the other's, it implies that the function is increasing at a faster rate. This comparison forms the basis for analyzing the overall behavior of the functions over an interval.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus connects differentiation and integration, allowing us to express the difference between function values as the integral of their derivatives. This is crucial when determining how a consistent difference in derivatives translates into a difference in the function values over an interval.

Initial Conditions

Initial conditions provide a specific reference point that can be used to determine the relative positions of functions. When two functions share an equal value at an initial point, any subsequent consistent difference in their derivatives will cause the functions to diverge in a predictable manner, as dictated by the accumulation of differences over an interval.

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