Animated Solution for Mathematics - Differentiation: The number of distinct real roots of the equation $3x^4+4x^3-12x^2+4=0$ is .

A Journey Through the Quartic

Welcome, fellow traveler. Today, we are not just solving an equation; we are embarking on a journey to visualize the hidden geometry of a quartic polynomial.

When you see an equation like $3x^4+4x^3-12x^2+4=0$, your first instinct might be to panic or start hunting for factors using the Rational Root Theorem. But I want you to pause. Take a breath.

In the world of JEE Advanced, the most powerful weapon is not brute force calculation—it is the ability to see the function as a living, breathing entity.

Defining the Function

Let us define our function as $f(x)=3x^4+4x^3-12x^2+4$. Our goal is to find the number of distinct real roots.

In geometric terms, this is equivalent to asking: "How many times does this curve cross the x-axis?" To answer this, we need to understand the 'landscape' of the function.

Where does it climb? Where does it dip? Where does it turn? This is where the magic of calculus comes into play.

Using the Derivative

To understand the turning points, we must look at the rate of change. We calculate the derivative, $f^{\prime }(x)$, which tells us the slope of the tangent at any point.

Using the power rule, we find:

Now, look at this expression. It is a cubic polynomial. To find the critical points—the peaks and valleys—we set $f^{\prime }(x)=0$.

Factoring this is a beautiful exercise in pattern recognition. We pull out the common term $12x$:

And the quadratic part factors perfectly into $(x+2)(x-1)$. Thus, our derivative is:

This tells us that the slope is zero at $x=-2$, $x=0$, and $x=1$. These are the three critical points where our curve changes direction.

Imagine standing on a roller coaster; these are the moments where you are momentarily weightless before the track dips or rises.

Evaluating the Terrain

Now, let us find the altitude of our roller coaster at these critical points. We substitute these values back into our original function $f(x)$:

1. At $x=-2$: We are deep in a valley.

2. At $x=0$: We have climbed to a local peak.

3. At $x=1$: We have dipped into another small valley.

Counting the Crossings

Now, let us connect the dots. We know the end behavior of a quartic with a positive leading coefficient: as $x\to \infty$, $f(x)\to \infty$, and as $x\to -\infty$, $f(x)\to \infty$.

Let us trace the path:

- From $-\infty$ to $-2$, the function comes down from $+\infty$ and reaches $-28$. It must cross the x-axis once. (Root 1)

- From $-2$ to $0$, the function climbs from $-28$ to $4$. It must cross the x-axis again. (Root 2)

- From $0$ to $1$, the function drops from $4$ to $-1$. It crosses the x-axis a third time. (Root 3)

- From $1$ to $+\infty$, the function rises from $-1$ to $+\infty$. It crosses the x-axis a fourth time. (Root 4)

By the Intermediate Value Theorem, because the function is continuous and changes sign in each of these intervals, it must cross the axis.

We have found 4 distinct real roots. We didn't need to solve the quartic; we simply mapped its journey. This is the elegance of calculus. Keep this perspective, and no polynomial will ever intimidate you again.