The function `f(x) ={{:( 5x-4, " for " 0 lt x le 1) ,( 4x^(2) - 3x, " for" 1 lt x lt 2 ),( 3x + 4 , "for" x ge2):}is `

Options

A. continuous at x-1 and x=2
B. continuous at x=1 but not derivable at x=2
C. continuous at x=1 and 2 but not derivable at x=1
D. continuous at x=1 and 2 but not derivable derivable at x=1 and x=2

Answer

To determine the continuity and differentiability of the piecewise function

\[ f(x) = \begin{cases} 5x - 4 & \text{for } 0 < x \leq 1 \\ 4x^2 - 3x & \text{for } 1 < x < 2 \\ 3x + 4 & \text{for } x \geq 2 \end{cases} \]

we need to check the points \(x = 1\) and \(x = 2\).

### Step 1: Check Continuity at \(x = 1\)

1. **Calculate \(f(1)\)**:
\[ f(1) = 5(1) - 4 = 1 \]

2. **Calculate \(\lim_{x \to 1^-} f(x)\)**:
For \(x < 1\), we use \(f(x) = 5x - 4\):
\[ \lim_{x \to 1^-} f(x) = 5(1) - 4 = 1 \]

3. **Calculate \(\lim_{x \to 1^+} f(x)\)**:
For \(x > 1\), we use \(f(x) = 4x^2 - 3x\):
\[ \lim_{x \to 1^+} f(x) = 4(1)^2 - 3(1) = 4 - 3 = 1 \]

4. **Check continuity**:
Since \(f(1) = \lim_{x \to 1^-} f(x) = \lim_{x \to 1^+} f(x) = 1\), \(f(x)\) is continuous at \(x = 1\).

### Step 2: Check Differentiability at \(x = 1\)

1. **Calculate \(f'(x)\) for \(0 < x < 1\)**:
\[ f'(x) = \frac{d}{dx}(5x - 4) = 5 \]

2. **Calculate \(f'(x)\) for \(1 < x < 2\)**:
\[ f'(x) = \frac{d}{dx}(4x^2 - 3x) = 8x - 3 \]
Evaluating at \(x = 1\):
\[ f'(1) = 8(1) - 3 = 5 \]

3. **Check differentiability**:
Since \(f'(1^-) = 5\) and \(f'(1^+) = 5\), \(f(x)\) is differentiable at \(x = 1\).

### Step 3: Check Continuity at \(x = 2\)

1. **Calculate \(f(2)\)**:
\[ f(2) = 3(2) + 4 = 6 + 4 = 10 \]

2. **Calculate \(\lim_{x \to 2^-} f(x)\)**:
For \(x < 2\), we use \(f(x) = 4x^2 - 3x\):
\[ \lim_{x \to 2^-} f(x) = 4(2)^2 - 3(2) = 16 - 6 = 10 \]

3. **Calculate \(\lim_{x \to 2^+} f(x)\)**:
For \(x > 2\), we use \(f(x) = 3x + 4\):
\[ \lim_{x \to 2^+} f(x) = 3(2) + 4 = 6 + 4 = 10 \]

4. **Check continuity**:
Since \(f(2) = \lim_{x \to 2^-} f(x) = \lim_{x \to 2^+} f(x) = 10\), \(f(x)\) is continuous at \(x = 2\).

### Step 4: Check Differentiability at \(x = 2\)

1. **Calculate \(f'(x)\) for \(1 < x < 2\)**:
\[ f'(x) = 8x - 3 \]
Evaluating at \(x = 2\):
\[ f'(2) = 8(2) - 3 = 16 - 3 = 13 \]

2. **Calculate \(f'(x)\) for \(x \geq 2\)**:
\[ f'(x) = 3 \]

3. **Check differentiability**:
Since \(f'(2^-) = 13\) and \(f'(2^+) = 3\), \(f(x)\) is not differentiable at \(x = 2\).

### Conclusion

- The function \(f(x)\) is continuous at \(x = 1\) and \(x = 2\).
- The function \(f(x)\) is differentiable at \(x = 1\) but not differentiable at \(x = 2\).