Animated Solution for Mathematics - Differentiation: How many real solutions does the equation $x^7+14x^5+16x^3+30x-560=0$ have?

Welcome, future engineer. Today, we are going to stare down a mathematical beast. Look at this equation:

At first glance, it looks terrifying. A degree-seven polynomial? How are we supposed to find the roots? Are we going to use synthetic division? Are we going to guess and check?

Stop. Take a deep breath. In the world of JEE Advanced, the most complex-looking problems often have the most elegant, simple solutions. We are not here to solve for $x$ using brute force; we are here to understand the soul of the function.

Let us define our function as $f(x)=x^7+14x^5+16x^3+30x-560$. When we ask for the number of real solutions, we are essentially asking: "How many times does this curve cross the $x$-axis?"

Imagine the graph of this function. It is a continuous, smooth line. It does not break, and it does not jump; it just flows. If we can figure out the general shape of this flow, we can answer the question without ever finding the specific value of $x$.

To understand the shape, we need to know if the function is climbing or falling. This is where the derivative, $f^{\prime }(x)$, becomes our best friend. The derivative tells us the slope of the tangent line at any point.

If the slope is positive, the function is climbing. If it is negative, it is falling. Let us differentiate $f(x)$ term by term using the power rule, $\frac{d}{dx}(x^n)=n{x}^{n-1}$.

Differentiating $x^7$ gives us $7x^6$. Differentiating $14x^5$ gives us $14\cdot 5x^4=70x^4$. Differentiating $16x^3$ gives us $16\cdot 3x^2=48x^2$. Differentiating $30x$ gives us $30$, and the constant $-560$ has a derivative of $0$.

Putting it all together, we get:

Now, look closely at this derivative. Notice anything? Every single power of $x$ is even: $x^6$, $x^4$, and $x^2$.

Why does this matter? Because any real number raised to an even power is always non-negative. Thus, $x^6\ge 0$, $x^4\ge 0$, and $x^2\ge 0$.

This means that $7x^6+70x^4+48x^2$ is always greater than or equal to zero. When we add $30$, we find that:

The derivative is always positive! It is never zero, and it is certainly never negative. This is a massive realization. It means our function $f(x)$ is strictly increasing. It never turns back and never wiggles; it is a relentless climb from the depths of negative infinity to the heights of positive infinity.

Because $f(x)$ is a polynomial, it is continuous. 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 goes from $-\infty$ to $+\infty$ must cross the $x$-axis at least once. But because we proved it is strictly increasing, it can never turn around to cross the axis a second time.

It crosses exactly once. And there you have it. We didn't need to solve for $x$ or factor the polynomial. We just needed to understand the behavior of the function. This is the beauty of calculus. It allows us to see the truth of a problem without getting lost in the arithmetic. Keep this mindset, and you will conquer any problem the JEE throws at you.