To solve this problem, we need to establish the conditions under which the function decreases and then increases. This typically implies that the function has a minimum point.

Let's break down the problem step by step:

### 1. Break Down the Problem
1. We know that $ f(x) $ decreases to a minimum point and then increases. This implies that the minimum point occurs at some $ x = c $, where $ f'(c) = 0 $.
2. We are given that $ f(1) = -29 $.

### 2. Relevant Concepts
For a function to have a local minimum:
- The first derivative $ f'(x) $ should change from negative to positive at the minimum point $ c $.
- Therefore, there should be critical points where $ f'(c) = 0 $.

Without loss of generality, we can assume $ f(x) $ is a quadratic function of the form:
$$
f(x) = ax^2 + bx + c
$$

### 3. Analysis and Detail
Given:
- The function decreases and then increases, which means $ a 0 $.
- $ f(1) = -29 $.

Substituting $ x = 1 $ into the quadratic function:
$$
f(1) = a(1)^2 + b(1) + c = a + b + c = -29 \quad (1)
$$

We also know that for the function to have a minimum, the vertex of the parabola (which can be found using the formula $ x = -\frac{b}{2a} $) must be at $ x = c $. However, without a specific location for $ c $, we need more information about $ f $ to find explicit values for $ a $ and $ b $.

### 4. Verify and Summarize
To find specific values for $ a $ and $ b $, we need either another point or additional conditions regarding $ f(x) $. The present information is insufficient to uniquely identify $ a $ and $ b $.

Thus, we have one equation (1) and no additional information to find the unique values of $ a $ and $ b $. To summarize:
- We established the function characteristics.
- We formulated a condition based on provided function value.
- However, we lack sufficient conditions to solve for $ a $ and $ b $ definitively.

### Final Answer
The values of $ a $ and $ b $ cannot be uniquely determined with the information provided. We need additional details about the function or another value of $ f(x) $ to solve for $ a $ and $ b $.

Question

To solve this problem, we need to establish the conditions under which the function decreases and then increases. This typically implies that the function has a minimum point.

Let's break down the problem step by step:

### 1. Break Down the Problem
1. We know that $ f(x) $ decreases to a minimum point and then increases. This implies that the minimum point occurs at some $ x = c $, where $ f'(c) = 0 $.
2. We are given that $ f(1) = -29 $.

### 2. Relevant Concepts
For a function to have a local minimum:
- The first derivative $ f'(x) $ should change from negative to positive at the minimum point $ c $.
- Therefore, there should be critical points where $ f'(c) = 0 $.

Without loss of generality, we can assume $ f(x) $ is a quadratic function of the form:
$$
f(x) = ax^2 + bx + c
$$

### 3. Analysis and Detail
Given:
- The function decreases and then increases, which means $ a > 0 $.
- $ f(1) = -29 $.

Substituting $ x = 1 $ into the quadratic function:
$$
f(1) = a(1)^2 + b(1) + c = a + b + c = -29 \quad (1)
$$

We also know that for the function to have a minimum, the vertex of the parabola (which can be found using the formula $ x = -\frac{b}{2a} $) must be at $ x = c $. However, without a specific location for $ c $, we need more information about $ f $ to find explicit values for $ a $ and $ b $.

### 4. Verify and Summarize
To find specific values for $ a $ and $ b $, we need either another point or additional conditions regarding $ f(x) $. The present information is insufficient to uniquely identify $ a $ and $ b $.

Thus, we have one equation (1) and no additional information to find the unique values of $ a $ and $ b $. To summarize:
- We established the function characteristics.
- We formulated a condition based on provided function value.
- However, we lack sufficient conditions to solve for $ a $ and $ b $ definitively.

### Final Answer
The values of $ a $ and $ b $ cannot be uniquely determined with the information provided. We need additional details about the function or another value of $ f(x) $ to solve for $ a $ and $ b $.

Knowee AI · Accepted Answer