Let `f(x) = (ax +b)/(cx + d)` `(da-cb ne 0, c ne0)` then f(x) has

Options

A. a critical point
B. no point of inflection
C. a maximum
D. a minimum

Answer

To solve the problem, we are given the function:

\[ f(x) = \frac{ax + b}{cx + d} \]

where \(da - cb \neq 0\) and \(c \neq 0\). We need to determine whether \(f(x)\) has a critical point, a point of inflection, or maximum/minimum values.

### Step 1: Find the derivative \(f'(x)\)

Using the quotient rule for differentiation, which states that if \(f(x) = \frac{g(x)}{h(x)}\), then:

\[ f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{(h(x))^2} \]

Here, \(g(x) = ax + b\) and \(h(x) = cx + d\).

- The derivative \(g'(x) = a\)
- The derivative \(h'(x) = c\)

Applying the quotient rule:

\[ f'(x) = \frac{(a)(cx + d) - (ax + b)(c)}{(cx + d)^2} \]

### Step 2: Simplify the derivative

Now, we simplify the numerator:

\[ f'(x) = \frac{acx + ad - (acx + bc)}{(cx + d)^2} \]

The \(acx\) terms cancel out:

\[ f'(x) = \frac{ad - bc}{(cx + d)^2} \]

### Step 3: Analyze the derivative

We know that \(da - cb \neq 0\), which means \(ad - bc \neq 0\). This implies that \(f'(x)\) is never equal to zero because the numerator \(ad - bc\) is a non-zero constant.

### Step 4: Determine critical points

Critical points occur where \(f'(x) = 0\) or \(f'(x)\) is undefined. Since \(f'(x)\) is never zero, there are no critical points.

### Step 5: Check for points of inflection

To find points of inflection, we need to check the second derivative \(f''(x)\). However, since \(f'(x)\) does not equal zero and is a constant divided by a square (which is always positive for real \(x\)), \(f'(x)\) does not change sign. Therefore, there are no points of inflection.

### Conclusion

Since \(f'(x)\) is never zero and does not change sign, we conclude that:

- There are no critical points.
- There are no points of inflection.
- There are no maximum or minimum values.

Thus, the correct answer is that \(f(x)\) has **no critical points, no points of inflection, and no maximum or minimum values**.