Let f(x) and g(x) be defined and differentiable for x ≥x0 and f(x0)=g(x0), f^'(x)>g^'(x) for x>x

step 1 We are given that f x 0 is equal to z x 0 and f das x is greater than z x for all x belong x gre

Let f(x) and g(x) be defined and differentiable for x ≥x0...

Let $f(x)$ and $g(x)$ be defined and differentiable for $x \geq x_{0}$ and $f\left(x_{0}\right)=g\left(x_{0}\right), f^{\prime}(x)>g^{\prime}(x)$ for $x>x_{0}$, then
(A) $f(x)<g(x), x>x_{0}$
(b) $f(x)=g(x), x>x_{0}$
(C) $f(x)>g(x), x>x_{0}$
(d) None of these

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Key Concepts

Initial Condition Influence

An initial condition where two functions are equal provides a baseline comparison. If one function's derivative is greater than the other's past this point, the initial equality combined with the differing rates of change ensures that the function with the larger derivative will eventually be greater beyond that initial point.

Mean Value Theorem

The Mean Value Theorem connects the average rate of change of a function on an interval with its instantaneous rate of change at some point within that interval. When two functions share the same initial value and one function has a higher derivative, the Mean Value Theorem helps demonstrate that the function with the higher derivative will exceed the other as the input increases.

Differentiability

Differentiability guarantees that functions have well-defined derivatives. This is crucial for analyzing how functions change and for applying theorems that relate a function’s derivative to its overall behavior.

Derivative Comparison

Comparing the derivatives of two functions provides insight into their rates of change. When one function consistently has a greater derivative than another on an interval, it implies that the function increases faster, which can determine the ordering of the functions on that interval.

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