f\left(x\right)=-\frac{\left(x-1\right)^2}{2}+3 What is the region of the domain where f(x) is decreasing?



To find the region where the function \( f(x) = -\frac{(x-1)^2}{2} + 3 \) is decreasing, we need to analyze the first derivative of the function and find where it is negative.

### Step 1: Find the first derivative of \( f(x) \).
The given function is:

\[
f(x) = -\frac{(x-1)^2}{2} + 3
\]

Now, we calculate the derivative \( f'(x) \):

\[
f'(x) = \frac{d}{dx} \left[ -\frac{(x-1)^2}{2} + 3 \right]
\]

Using the chain rule, we get:

\[
f'(x) = -\frac{2(x-1)}{2} = -(x-1)
\]

### Step 2: Analyze the sign of \( f'(x) \).
The derivative is \( f'(x) = -(x-1) \). To determine where \( f(x) \) is decreasing, we need to find where \( f'(x) < 0 \):

\[
-(x-1) < 0
\]

Solving this inequality:

\[
x-1 > 0 \quad \Rightarrow \quad x > 1
\]

### Conclusion:
The function \( f(x) \) is **decreasing** for \( x > 1 \).

Would you like further details or have any questions?

---

Here are 5 related questions:
1. What is the region where \( f(x) \) is increasing?
2. How do we find critical points of a function?
3. What is the maximum or minimum value of \( f(x) \)?
4. How do second derivatives help in determining concavity?
5. Can you sketch the graph of \( f(x) \)?

**Tip:** The sign of the first derivative gives important information about where a function is increasing or decreasing!

f(x) = -\frac{(x-1)^2}{2} + 3 What is the region of the domain where f(x) is decreasing?

Explore how to find the region where the function f(x) = -\frac{(x-1)^2}{2} + 3 is decreasing using the first derivative. This solution covers the concept of derivatives and the first derivative test, suitable for Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation