The absolute maximum value of $\mathrm{y}={\mathrm{x}}^3-3\mathrm{x}+2 \mathrm{in} 0\le \mathrm{x}\le 2 \mathrm{is}$

Explanation:

We need to find the absolute maximum value of the function:

over the interval 0 ≤ x ≤ 2.

Step 1: Differentiate the Function

To find the critical points, take the derivative:

Step 2: Set the Derivative Equal to Zero

Setting the derivative equal to zero:

Since our interval is 0 ≤ x ≤ 2, we consider only x = 1.

Step 3: Evaluate the Function at Critical Points and Boundaries

Evaluating the function:

Step 4: Identify the Maximum

The maximum value occurs at:

Final Answer: