Animated Solution for Mathematics - Differentiation: Given $P(x)=x^4+a{x}^3+b{x}^2+cx+d$ such that $x=0$ is the only real root of $P^{\prime }(x)=0$. If $P(-1)<P(1)$, then in the interval $[-1,1]$:

Imagine you are standing before a grand, sweeping landscape—a mathematical curve defined by the quartic polynomial:

It looks complex, perhaps even intimidating, but every curve has a story, and every story has a turning point. Our journey begins by seeking the derivative, the heartbeat of the function:

The problem whispers a secret: $x=0$ is the only real root of $P^{\prime }(x)=0$. This is our golden key. If we substitute $x=0$ into the derivative, we find $P^{\prime }(0)=c=0$. The constant $c$ vanishes, leaving us with:

Now, we must be careful. We are told $x=0$ is the only real root. This means the quadratic factor $4x^2+3ax+2b$ cannot have any real roots.

If it did, those roots would also be roots of $P^{\prime }(x)$, violating our condition. Since the leading coefficient $4$ is positive, this quadratic must be strictly positive for all real $x$.

It never touches the x-axis; it floats above it like a bird in flight. This is a profound realization: the sign of $P^{\prime }(x)$ is entirely controlled by the factor $x$.

Let us observe the behavior of $P(x)$ on the interval $[-1,1]$. When $x<0$, $P^{\prime }(x)$ is negative, meaning our function is sliding downhill, strictly decreasing.

When $x>0$, $P^{\prime }(x)$ is positive, and the function climbs, strictly increasing. At $x=0$, the function reaches its lowest point—a local minimum.

Now, consider the boundaries. We are given $P(-1)<P(1)$. Since the function decreases from $-1$ to $0$ and increases from $0$ to $1$, we know $P(0)$ is the minimum.

The maximum must occur at one of the endpoints, either $x=-1$ or $x=1$. Because $P(1)>P(-1)$, the highest point in our interval is undeniably $P(1)$.

We have navigated the slopes and valleys of this quartic. We found that $P(0)$ is the minimum, and $P(1)$ is the maximum.

Thus, $P(-1)$ is not the minimum, and $P(1)$ is the maximum. You have successfully decoded the geometry of the polynomial.