Questions: (b) Find the maximum or minimum value of f. (c) Find the domain and range of f. (Enter your answers using interval notation.) domain (-∞, ∞) range

Solution Steps

To find the maximum or minimum value of a function \( f(x) \), we need to determine the critical points by taking the derivative of the function and setting it to zero. Then, we evaluate the function at these critical points to find the maximum or minimum values. For the domain and range, we analyze the function's behavior and constraints.

To find the maximum or minimum value of \( f \) and its domain and range, we first need to identify the function \( f(x) \). However, the function \( f(x) \) is not provided in the question. For the sake of this example, let's assume \( f(x) = x^2 - 4x + 3 \).

To find the maximum or minimum value of a quadratic function \( f(x) = ax^2 + bx + c \), we use the vertex formula \( x = -\frac{b}{2a} \).

For \( f(x) = x^2 - 4x + 3 \):

• \( a = 1 \)
• \( b = -4 \)
• \( c = 3 \)

The vertex \( x \) is: \[ x = -\frac{-4}{2 \cdot 1} = \frac{4}{2} = 2 \]

Now, substitute \( x = 2 \) back into the function to find \( f(2) \): \[ f(2) = (2)^2 - 4(2) + 3 = 4 - 8 + 3 = -1 \]

Since \( a > 0 \), the parabola opens upwards, and the vertex represents the minimum value.

The domain of any quadratic function is all real numbers, \( (-\infty, \infty) \).

The range of \( f(x) = x^2 - 4x + 3 \) starts from the minimum value \( -1 \) and goes to \( \infty \): \[ \text{Range} = [-1, \infty) \]

Final Answer