JAMIA 2024 — Mathematics PYQ

Step 1: Find the Derivative

The given function is:

Differentiating with respect to $x$:

Step 2: Find Critical Points

Set the derivative equal to zero to find the critical points:

Since our interval is $0\le x\le 2$, we only consider $x=1$. The point $x=-1$ is outside the given range.

Step 3: Evaluate the Function at Key Points

We now calculate the value of $y$ at the critical point ($x=1$) and the boundaries ($x=0$ and $x=2$):

1. At $x=0$ (Lower Bound):

   $$f(0)=(0{)}^3-3(0)+2=2$$

2. At $x=1$ (Critical Point):

   $$f(1)=(1{)}^3-3(1)+2=1-3+2=0$$

3. At $x=2$ (Upper Bound):

   $$f(2)=(2{)}^3-3(2)+2=8-6+2=4$$

Step 4: Compare the Values

Comparing the results:

• $f(0)=2$

• $f(1)=0$

• $f(2)=4$

The largest value among these is 4.

Final Answer

The absolute maximum value of $y=x^3-3x+2$ in the interval $[0,2]$ is: