Animated Solution for Mathematics - Differentiation: The number of distinct real roots of $x^4-4x+1=0$ is :

The function defining our path is given by:

Our objective is to determine the number of times this curve intersects the $x$-axis, which corresponds to finding the number of real roots of the equation $f(x)=0$.

To understand the behavior of the curve, we examine its slope using the first derivative:

Setting the derivative to zero identifies the critical points where the slope is horizontal:

This reveals that the function has only one real critical point at $x=1$. Consequently, the curve does not oscillate; it descends to a single minimum and then rises indefinitely.

We determine the depth of this valley by evaluating the function at the critical point $x=1$:

The minimum point of the landscape is located at $(1,-2)$. Since this minimum lies below the $x$-axis, the curve must cross the axis once while descending and once while ascending.

We apply the Intermediate Value Theorem to confirm the existence of these roots. First, consider the interval $[0,1]$:

Because the function is continuous, there must exist at least one root $\alpha$ in the interval $(0,1)$. Next, consider the interval $[1,2]$:

Because the function transitions from negative to positive, there must exist at least one root $\beta$ in the interval $(1,2)$.

We have identified two distinct roots, $\alpha$ and $\beta$. To determine if others exist, we recall that the function is strictly decreasing for $x<1$ and strictly increasing for $x>1$.

A function that is strictly monotonic on an interval can cross the $x$-axis at most once within that interval. Therefore, no additional roots can exist.

The landscape crosses the $x$-axis exactly twice.