Animated Solution for Mathematics - Differentiation: Let M and m be respectively the absolute maximum and the absolute minimum value of the function, $f(x)=2x^3-9x^2+12x+5$ in the interval [0,3]. Then $M-m$ is equal to :

Imagine you are standing at the start of a roller coaster track. The track is defined by the function $f(x)=2x^3-9x^2+12x+5$, and your ride is strictly limited to the section between $x=0$ and $x=3$.

Your goal is to find the highest peak ($M$) and the lowest valley ($m$) on this specific stretch of track. This is the essence of the Extreme Value Theorem, which states that for any continuous function on a closed interval, an absolute maximum and minimum must exist.

To find these extrema, we must examine two types of locations: the boundaries of our interval and the points where the track levels off. The boundaries are simply $x=0$ and $x=3$.

To find the "level" points, we calculate the derivative of the function to determine where the slope of the tangent line is zero:

Setting this derivative to zero identifies the critical points where the roller coaster is momentarily flat:

Dividing by $6$, we obtain the simplified quadratic equation $x^2-3x+2=0$. Factoring this yields $(x-1)(x-2)=0$, providing critical points at $x=1$ and $x=2$.

Since both $x=1$ and $x=2$ fall within our interval $[0,3]$, they are both valid candidates for our search.

We now perform a final audit by evaluating the original function $f(x)$ at each of our four candidates: the boundaries ($x=0,3$) and the critical points ($x=1,2$).

1. At $x=0$:

2. At $x=1$:

3. At $x=2$:

4. At $x=3$:

Comparing the results $\{5,10,9,14\}$, the absolute maximum $M$ is $14$ (at $x=3$), and the absolute minimum $m$ is $5$ (at $x=0$).

The objective is to find the difference $M-m$. Using our determined values:

By systematically checking the boundaries and the critical points, we have successfully navigated the roller coaster of this cubic function. The final result is $9$.