Animated Solution for Mathematics - Differentiation: If Rolle's theorem holds for the function $f(x)=x^3-a{x}^2+bx-4,x\in [1,2]$ with $f^{\prime }(\frac{4}{3})=0$, then ordered pair $(a,b)$ is equal to :

Imagine you are standing on a mountain path. You start at an elevation of $1000$ meters at point $A$, and after a long, winding hike, you find yourself back at an elevation of $1000$ meters at point $B$.

Logic dictates that at some point during your journey, you must have either climbed up and then back down, or descended and then climbed back up. In either case, there was a moment where your path was perfectly level—a horizontal tangent. This is the physical soul of Rolle's Theorem.

We are given the cubic function $f(x)=x^3-a{x}^2+bx-4$ on the interval $[1,2]$. For Rolle's Theorem to hold, we require two fundamental conditions.

First, the function must be continuous and differentiable (which, being a polynomial, it is). Second, the values at the boundaries must be identical: $f(1)=f(2)$.

Let us calculate these values. Substituting $x=1$ into our function, we get:

Now, for $x=2$, we have:

Setting these equal, we arrive at our first vital equation:

Now, we turn to the second condition: the existence of a horizontal tangent. We are told that $f^{\prime }(\frac{4}{3})=0$.

To find this, we differentiate our function:

By substituting $x=\frac{4}{3}$ into this derivative, we get:

Simplifying this, we have $3(\frac{16}{9})-\frac{8a}{3}+b=0$, which reduces to $\frac{16}{3}-\frac{8a}{3}+b=0$. Multiplying by $3$ to clear the fractions, we obtain our second equation:

We are left with a system of two linear equations:

To solve this, we multiply the first equation by $3$ to get $9a-3b=21$. Now, subtracting the second equation from this, we see the $b$ terms vanish:

This calculation yields $a=5$. Substituting $a=5$ back into our first equation, $3(5)-b=7$, we find $15-b=7$, leading to $b=8$.

We have successfully navigated the constraints of the theorem to find that $(a,b)=(5,8)$. It is a beautiful reminder that even in the abstract world of calculus, every equation has a physical story to tell.