Animated Solution for Mathematics - Differentiation: Let $f:(a,b)\to \mathbb{R}$ be twice differentiable function such that $f(x)={\int }_a^{x}g(t)dt$ for a differentiable function $g(x)$. If $f(x)=0$ has exactly five distinct roots in $(a,b)$, then $g(x)g^{\prime }(x)=0$ has at least :

Imagine you are standing on a vast, rolling landscape looking at the graph of a function $f(x)$. You know that this function crosses the ground—the $x$-axis—exactly five times. These five points are your anchors, denoted as $x_1,x_2,x_3,x_4,x_5$.

As a mathematician, you know that a function is not just a static line; it is a story of motion. To get from one root to the next, the function must climb, peak, and descend. It is in this "climbing and descending" that the magic of calculus reveals itself.

We are given the relationship $f(x)={\int }_a^{x}g(t)dt$. This is the Fundamental Theorem of Calculus in its purest form, representing $f(x)$ as the accumulation of $g(t)$.

If we want to know how $f(x)$ is changing at any moment, we simply look at $g(x)$. Mathematically, we differentiate both sides:

This is our bridge. Every time $f(x)$ reaches a peak or a valley—a point where it stops moving up and starts moving down—the slope $f^{\prime }(x)$ must be zero. Since $f^{\prime }(x)=g(x)$, this implies that $g(x)$ must be zero at those exact moments.

Now, let us apply the elegant logic of Rolle's Theorem. If $f(x)$ has five distinct roots, it must have at least four "turning points" between them.

Between $x_1$ and $x_2$, there is a peak or valley; between $x_2$ and $x_3$, another, and so on. Since there are four such intervals, $f^{\prime }(x)$—and therefore $g(x)$—must have at least four roots. Let us call these roots $y_1,y_2,y_3,y_4$.

Think of $g(x)$ as a new function. We just established that $g(x)$ has at least four roots: $y_1,y_2,y_3,y_4$.

If we apply Rolle's Theorem to $g(x)$, we look at the intervals between its roots: $(y_1,y_2)$, $(y_2,y_3)$, and $(y_3,y_4)$. In each of these three intervals, $g(x)$ must have a turning point. A turning point of $g(x)$ is simply a point where its derivative, $g^{\prime }(x)$, is zero.

Therefore, $g^{\prime }(x)$ must have at least three roots, which we can call $z_1,z_2,z_3$. Geometrically, these are the inflection points of our original function $f(x)$, where the curvature changes from concave up to concave down.

We are looking for the roots of the product $g(x)g^{\prime }(x)=0$. This product is zero if either $g(x)=0$ or $g^{\prime }(x)=0$.

We have identified: 1. At least 4 roots from $g(x)=0$. 2. At least 3 roots from $g^{\prime }(x)=0$.

Because the roots of $g^{\prime }(x)$ are strictly interlaced between the roots of $g(x)$, these two sets of roots are entirely distinct and do not overlap. Thus, the total number of roots for the product is simply the sum of the two sets:

It is a beautiful result. By simply knowing the number of times a function crosses the axis, we have peered into the behavior of its derivative and its second derivative, uncovering a hidden structure of at least seven roots. You have successfully navigated the landscape of the function, from its roots to its peaks, and finally to its inflection points.