Animated Solution for Mathematics - Differentiation: The sum of the absolute maximum and minimum values of the function $f(x)=|x^2-5x+6|-3x+2$ in the interval $[-1,3]$ is equal to :

The function $f(x)=|x^2-5x+6|-3x+2$ is a mathematical chameleon. The modulus bars act as a gatekeeper, changing the function's behavior at critical moments.

Our goal is to find the sum of the absolute maximum and minimum values on the interval $[-1,3]$.

The first step is to examine the heart of the expression: $x^2-5x+6$. We factorize this to find the roots:

These roots, $x=2$ and $x=3$, are the points where the expression inside the modulus changes sign. This is the crucial realization: the function behaves differently depending on whether the quadratic is positive or negative.

Because the behavior changes at $x=2$, we must split the interval $[-1,3]$ into two distinct regions: $[-1,2]$ and $[2,3]$.

In the first region, $[-1,2]$, the quadratic $(x-2)(x-3)$ is non-negative. Thus, the modulus opens positively:

In the second region, $[2,3]$, the quadratic is non-positive. Here, the modulus opens with a negative sign:

Now, let us analyze the nature of these two pieces. By calculating the derivative for the first piece, $f^{\prime }(x)=2x-8$, we see that for any $x\le 2$, the derivative is always negative.

Similarly, for the second piece, $f^{\prime }(x)=-2x+2$, we see that for any $x\ge 2$, the derivative is also negative. This is a beautiful result: the function is strictly decreasing across the entire interval $[-1,3]$.

Because the function is strictly decreasing, the absolute maximum must occur at the start of our interval, $x=-1$. Plugging this into our first piece:

The absolute minimum must occur at the end of our interval, $x=3$. Plugging this into our second piece:

The sum of these values is $17+(-7)=10$. We have navigated the modulus and arrived at the final answer of 10.