Animated Solution for Mathematics - Differentiation: Let $p(x)$ be a real polynomial of least degree which has a local maximum at $x=1$ and a local minimum at $x=3$. If $p(1)=6$ and $p(3)=2$, then $p^{\prime }(0)$ is

Imagine you are standing in front of a graph. You see a smooth, elegant curve—a polynomial—that climbs to a majestic peak at $x=1$ and then descends into a deep valley at $x=3$.

We are given two critical anchors: a local maximum at $(1,6)$ and a local minimum at $(3,2)$. In the language of calculus, these points are moments of stillness where the tangent line is perfectly horizontal.

Mathematically, this translates to the derivative being zero at these points. Our first insight is that $p^{\prime }(1)=0$ and $p^{\prime }(3)=0$.

We seek the polynomial of the least degree. Since the derivative $p^{\prime }(x)$ is zero at two distinct points, it must have at least two roots.

The simplest polynomial that has two roots is a quadratic. Thus, we can model the derivative as:

Here, $k$ is a scaling constant that determines the overall steepness of our curve. Expanding this, we get:

This quadratic expression is the engine of our polynomial; it dictates the rate of change at every point along the $x$-axis.

To find the original function $p(x)$, we must reverse the process of differentiation by integrating the derivative:

Performing this integration gives us:

Here, $C$ is our constant of integration, the vertical anchor that allows us to slide the entire curve up or down until it hits our target points.

We have two unknowns, $k$ and $C$, and two pieces of information: $p(3)=2$ and $p(1)=6$. Let's start with the local minimum at $x=3$.

Substituting $x=3$ into our equation, we find:

Notice the beautiful symmetry here: $9-18+9$ equals zero. This simplifies our equation to $C=2$.

With $C$ in hand, we turn to the local maximum at $x=1$. Substituting $x=1$ and $C=2$ into $p(1)=6$, we get:

Simplifying the bracket, we have $k(\frac{4}{3})=4$, which leads us directly to $k=3$.

We have successfully reconstructed the function. Our derivative is $p^{\prime }(x)=3(x-1)(x-3)$.

The question asks for the slope of the tangent at the $y$-axis, which is $p^{\prime }(0)$. Substituting $x=0$ into our derivative, we calculate:

The slope at the $y$-axis is $9$. Through this journey, we have seen how the geometry of a curve dictates its algebraic form, and how the power of calculus allows us to reconstruct the whole from the behavior of its parts.