Conceptual question on graphs

Question

I'm a student of economics but I have encountered a mathematical doubt while studying some material (nothing new!). I will try to present the question without any reference to economic concepts:

I have a curve which is shaped like a hill and a second curve on the same graph which is an upward sloping curve with a slight curvature at the start. The second curve is below the first one. The text states that the distance between the two curves is maximum at the point where the slopes of the two curves are equal. Why is that true?

Answer 1

Let the $2$ functions be $f(x)$ and $g(x)$. Then, we wish to find the point at which $f(x)-g(x)$ is maximum.

If we define $h(x) = f(x) - g(x)$, then the question becomes - find the $x$ at which $h(x)$ is maximum. Now, we know that this occurs at the maxima of the function, which are one of the points where $h'(x) = 0$, or where $f'(x) = g'(x)$. Thus, the distance between the $2$ functions is maximum when their slopes are equal.

Answer 2

Let $f$ and $g$ be continuous and smooth functions on some interval $I$ such that $f(x) \ge g(x)$ for all $x \in I$. Then, the "distance" between the curves on $I$ is yet another function $h$ given by $h(x)=f(x)-g(x)$. Finding the point of maximum distance is now just matter of finding the point where $h$ is at a maximum.

Now, if you are familiar with calculus, you know we can find this rather quickly by the first derivative test. The test states that if extrema (relative maximums or minimums) for some function $h$ exist on $I$, then they will occur at values of $x$ for which the derivative of $h$, or $h'(x)$, is equal to zero or undefined. Since the functions $f$ and $g$ are continuous and smooth on $I$, then $h$ is continuous and smooth on $I$, and so $h$ is defined for all $x$ on $I$. So then, our maximum must lie at values of $x$ for which $h'(x)=0$. Now, $h'(x)$ is found as follows:

$$\frac{d}{dx}h(x) = \frac{d}{dx} \bigg[ f(x)-g(x) \bigg]$$ $$\frac{d}{dx}h(x) = \frac{d}{dx} f(x) - \frac{d}{dx} g(x)$$ $$h'(x) = f'(x) - g'(x)$$

Now set $h'(x)=f'(x) - g'(x)=0$ and observe

$$0= f'(x) - g'(x)$$ $$g'(x) = f'(x)$$

Hence, the maximum of $h$ occurs wherever the $f'(x)$ and $g'(x)$ are equal. Of course, the derivative of a function conveys the slope of a curve at a single point on the curve, so this statement effectively tells us that the maximum of $h$, which tells us the maximum distance between curves $f$ and $g$, occurs at a value of $x$ such that the slopes of $f$ and $g$ are equal.