Animated Solution for Mathematics - Differentiation: Let $f:\mathbb{R}\to \mathbb{R}$ be a thrice differentiable function such that $f(0)=0,f(1)=1,f(2)=-1,f(3)=2$ and $f(4)=-2$. Then, the minimum number of zeros of $(3f^{\prime }{f}^{\prime \prime }+f{f}^{\prime \prime \prime })(x)$ is _______

We are given a thrice differentiable function $f(x)$ with the following values: $f(0)=0$, $f(1)=1$, $f(2)=-1$, $f(3)=2$, and $f(4)=-2$.

Our objective is to determine the minimum number of zeros for the expression $(3f^{\prime }{f}^{\prime \prime }+f{f}^{\prime \prime \prime })(x)$.

Recognizing the Pattern

At first glance, the expression $(3f^{\prime }{f}^{\prime \prime }+f{f}^{\prime \prime \prime })(x)$ appears complex. However, we can identify a "parent" function by observing the structure of its derivatives. Let us define a function $g(x)=f(x)f^{\prime }(x)$.

Applying the product rule to $g(x)$, we find the first derivative:

Differentiating once more to find $g^{\prime \prime }(x)$, we apply the product rule to both terms:

Combining the like terms, we arrive at the elegant result:

Thus, the expression we are investigating is exactly the second derivative of $g(x)=f(x)f^{\prime }(x)$.

The Intermediate Value Theorem

We analyze the roots of $f(x)$ using the Intermediate Value Theorem (IVT). We are given $f(0)=0$.

Between the given points, the function changes sign as follows: 1. In $(1,2)$, $f(x)$ changes from $1$ to $-1$, implying at least one zero. 2. In $(2,3)$, $f(x)$ changes from $-1$ to $2$, implying at least one zero. 3. In $(3,4)$, $f(x)$ changes from $2$ to $-2$, implying at least one zero.

Including the known zero at $x=0$, $f(x)$ has at least $4$ distinct zeros.

Rolle's Theorem

We now apply Rolle's Theorem to determine the zeros of the derivatives. If $f(x)$ has $4$ zeros, then $f^{\prime }(x)$ must have at least $4-1=3$ zeros, located between the consecutive zeros of $f(x)$.

Recall that $g(x)=f(x)f^{\prime }(x)$. The zeros of $g(x)$ occur whenever $f(x)=0$ or $f^{\prime }(x)=0$.

Since the zeros of $f(x)$ and $f^{\prime }(x)$ are distinct, $g(x)$ possesses at least $4+3=7$ zeros.

Reaching the Goal

We apply Rolle's Theorem iteratively to the function $g(x)$. Since $g(x)$ has at least $7$ zeros, its first derivative $g^{\prime }(x)$ must have at least $7-1=6$ zeros.

Applying Rolle's Theorem once more to $g^{\prime }(x)$, we conclude that $g^{\prime \prime }(x)$ must have at least $6-1=5$ zeros.

The expression $(3f^{\prime }{f}^{\prime \prime }+f{f}^{\prime \prime \prime })(x)$ is equivalent to $g^{\prime \prime }(x)$. Therefore, the minimum number of zeros for the given expression is 5.