Animated Solution for Mathematics - Differentiation: The number of real solutions of $x^7+5x^3+3x+1=0$ is equal to ________

Imagine you are standing before a mathematical mountain. The equation $f(x)=x^7+5x^3+3x+1=0$ looks like a jagged, intimidating peak.

If you try to solve this using standard algebraic methods, you will quickly find yourself lost in a labyrinth of impossible calculations. However, you do not need to find the exact value of $x$ to understand the nature of the roots. We are going to use the power of calculus to map the terrain of this function.

When algebra fails, calculus provides the map. We define our function as $f(x)=x^7+5x^3+3x+1$.

Our goal is to find how many times this function touches the $x$-axis, which is equivalent to finding the number of real solutions. To do this, we need to know if the function is climbing or falling by calculating the derivative, $f^{\prime }(x)$.

Using the power rule, where the derivative of $x^n$ is $n{x}^{n-1}$, we differentiate each term:

Now, look closely at this derivative. We have $7x^6+15x^2+3$.

Notice that $x^6$ and $x^2$ are even powers. For any real number $x$, an even power is always non-negative, meaning $x^6\ge 0$ and $x^2\ge 0$.

Consequently, $7x^6\ge 0$ and $15x^2\ge 0$. When we add the constant $3$ to these non-negative terms, the entire expression $f^{\prime }(x)$ must be at least $3$.

This is a profound realization: the derivative is strictly positive for all real $x$. In the language of calculus, this means our function $f(x)$ is strictly increasing. It is a path that only goes uphill.

Because the highest power of our polynomial is $7$ (an odd number), the end behavior is clear: as $x\to -\infty$, $f(x)\to -\infty$, and as $x\to \infty$, $f(x)\to \infty$.

By the Intermediate Value Theorem, a continuous function that travels from the depths of negative infinity to the heights of positive infinity must cross the $x$-axis at least once.

But because our function is strictly increasing, it can never turn back down. It crosses the $x$-axis and keeps climbing forever; it cannot cross again.

Therefore, there is exactly one real solution. You have successfully conquered the seventh-degree monster using nothing but the elegance of monotonicity.