For which value of x is the instantaneous rate of change in f(x) = -x*2 + x - 1 negative?
A) x= -2
B) x= -1
C) x= 1
D) x= 0

To determine the value of \( x \) where the instantaneous rate of change in the function \( f(x) = -x^2 + x - 1 \) is negative, we first need to find the derivative of the function, which represents the rate of change.

### Step 1: Find the derivative of \( f(x) \)
The function is given as:
\[
f(x) = -x^2 + x - 1
\]
Taking the derivative of \( f(x) \) with respect to \( x \):
\[
f'(x) = \frac{d}{dx}(-x^2 + x - 1) = -2x + 1
\]
This derivative, \( f'(x) = -2x + 1 \), represents the instantaneous rate of change of \( f(x) \).

### Step 2: Set the derivative less than zero (to find when the rate of change is negative)
We want to find the values of \( x \) where the rate of change is negative:
\[
f'(x) < 0
\]
Substitute the expression for \( f'(x) \):
\[
-2x + 1 < 0
\]
Solve for \( x \):
\[
-2x < -1
\]
\[
x > \frac{1}{2}
\]

### Step 3: Analyze the options
From the inequality \( x > \frac{1}{2} \), the rate of change is negative for values of \( x \) greater than \( \frac{1}{2} \).

Looking at the options:
- \( A) x = -2 \) is less than \( \frac{1}{2} \).
- \( B) x = -1 \) is also less than \( \frac{1}{2} \).
- \( C) x = 1 \) is greater than \( \frac{1}{2} \).
- \( D) x = 0 \) is less than \( \frac{1}{2} \).

The correct answer is \( \boxed{C} \), because at \( x = 1 \), the rate of change is negative.

Would you like more details on any step?

### Related Questions:
1. What does the derivative \( f'(x) \) tell us about the behavior of \( f(x) \)?
2. How do you interpret the sign of a derivative in terms of increasing and decreasing functions?
3. How would you solve a similar problem for higher-degree polynomials?
4. Can a function have points where the rate of change is zero?
5. What is the difference between the instantaneous rate of change and the average rate of change?

**Tip**: The derivative tells you whether a function is increasing or decreasing; when it's negative, the function is decreasing!

For which value of x is the instantaneous rate of change in f(x) = -x^2 + x - 1 negative?
A) x= -2
B) x= -1
C) x= 1
D) x= 0

Discover how to find the value of x where the instantaneous rate of change of the quadratic function f(x) = -x^2 + x - 1 is negative. This problem involves finding the derivative and solving inequalities, ideal for students in Grades 10-12.

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