If f(x) is continuous and differentiable in `x in [-7,0] and f'(x) le 2 AA x in [-7,0]`, also f(-7)=-3 then range of f(-1)+f(0)

Options

A. 1. `(-oo,11)`
B. 2. `[-6,20]`
C. 3. `(-oo, 20]`
D. 4. `[-3,11]`

Answer

To solve the problem, we need to analyze the function \( f(x) \) given the conditions provided. Let's break it down step by step.

### Step 1: Understand the Given Information
We know that:
- \( f(x) \) is continuous and differentiable on the interval \( [-7, 0] \).
- The derivative \( f'(x) \leq 2 \) for all \( x \in [-7, 0] \).
- The value \( f(-7) = -3 \).

### Step 2: Apply the Mean Value Theorem
According to the Mean Value Theorem, if \( f(x) \) is continuous on the closed interval and differentiable on the open interval, there exists at least one \( c \) in \( (a, b) \) such that:

\[ f'(c) = \frac{f(b) - f(a)}{b - a} \]

### Step 3: Analyze \( f(-1) \)
Let's apply the Mean Value Theorem on the interval \( [-7, -1] \):
- Let \( a = -7 \) and \( b = -1 \).

Using the Mean Value Theorem:

\[ f'(-c) = \frac{f(-1) - f(-7)}{-1 + 7} \quad \text{for some } c \in (-7, -1) \]

This simplifies to:

\[ f'(-c) = \frac{f(-1) + 3}{6} \]

Since \( f'(x) \leq 2 \):

\[ \frac{f(-1) + 3}{6} \leq 2 \]

Multiplying both sides by 6:

\[ f(-1) + 3 \leq 12 \]

Thus,

\[ f(-1) \leq 9 \]

### Step 4: Analyze \( f(0) \)
Now, let's apply the Mean Value Theorem on the interval \( [-7, 0] \):
- Let \( a = -7 \) and \( b = 0 \).

Using the Mean Value Theorem:

\[ f'(-c) = \frac{f(0) - f(-7)}{0 + 7} \quad \text{for some } c \in (-7, 0) \]

This simplifies to:

\[ f'(-c) = \frac{f(0) + 3}{7} \]

Since \( f'(x) \leq 2 \):

\[ \frac{f(0) + 3}{7} \leq 2 \]

Multiplying both sides by 7:

\[ f(0) + 3 \leq 14 \]

Thus,

\[ f(0) \leq 11 \]

### Step 5: Calculate \( f(-1) + f(0) \)
Now we have:

\[ f(-1) \leq 9 \quad \text{and} \quad f(0) \leq 11 \]

Adding these inequalities:

\[ f(-1) + f(0) \leq 9 + 11 = 20 \]

### Conclusion
Thus, the range of \( f(-1) + f(0) \) is \( (-\infty, 20] \).

### Final Answer
The range of \( f(-1) + f(0) \) is \( (-\infty, 20] \).

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