Draw the graph of function `f(x)={underset([x] " "1 le x le 2)(x " "0 le x le 1).` Graphically comment on the monotonic behaviour of f(x) at x=1 . Is f(x) M.I. for `X in [0,2]`

Answer

To solve the problem, we need to analyze the function defined as follows:

\[ f(x) = \begin{cases} x & \text{for } 0 \leq x < 1 \\ \lfloor x \rfloor & \text{for } 1 \leq x \leq 2 \end{cases} \]

### Step 1: Define the function piecewise
The function is defined in two parts:
- For \(0 \leq x < 1\), \(f(x) = x\).
- For \(1 \leq x < 2\), \(f(x) = \lfloor x \rfloor\) (the greatest integer less than or equal to \(x\)).

### Step 2: Evaluate the function at key points
- At \(x = 0\): \(f(0) = 0\)
- At \(x = 1\): \(f(1) = \lfloor 1 \rfloor = 1\)
- At \(x = 1.5\): \(f(1.5) = \lfloor 1.5 \rfloor = 1\)
- At \(x = 2\): \(f(2) = \lfloor 2 \rfloor = 2\)

### Step 3: Sketch the graph
1. For \(0 \leq x < 1\), plot the line \(y = x\) from \((0,0)\) to \((1,1)\).
2. At \(x = 1\), the function value is \(1\) (point included).
3. For \(1 < x < 2\), the function value remains constant at \(1\) until \(x = 2\), where it jumps to \(2\).
4. The graph will look like this:
- A line segment from \((0,0)\) to \((1,1)\).
- A horizontal line from \((1,1)\) to \((2,1)\).
- A point at \((2,2)\).

### Step 4: Analyze monotonic behavior at \(x = 1\)
To analyze the monotonic behavior at \(x = 1\):
- For \(x < 1\) (approaching from the left), \(f(1 - h) = 1 - h\) which is less than \(f(1) = 1\).
- For \(x > 1\) (approaching from the right), \(f(1 + h) = 1\) which is equal to \(f(1)\).

Thus, we can conclude that:
- \(f(1 - h) < f(1)\) and \(f(1 + h) = f(1)\).
- Therefore, \(f(x)\) is increasing at \(x = 1\).

### Step 5: Check if \(f(x)\) is monotonically increasing on \([0, 2]\)
- From \(0\) to \(1\), \(f(x)\) is increasing.
- From \(1\) to \(2\), \(f(x)\) is constant (not increasing).
- Hence, \(f(x)\) is not monotonically increasing on the interval \([0, 2]\).

### Final Conclusion
- The function \(f(x)\) is increasing at \(x = 1\).
- The function \(f(x)\) is not monotonically increasing on the interval \([0, 2]\).