The function f(x)=(4x^2-5x)/3 has an average rate of change of -2 over the interval [c,0], find the value of c.

We are given the function \( f(x) = \frac{4x^2 - 5x}{3} \) and asked to find the value of \( c \) such that the average rate of change of \( f(x) \) over the interval \([c, 0]\) is \(-2\).

The formula for the average rate of change of a function \( f(x) \) over the interval \([a, b]\) is:

\[
\text{Average rate of change} = \frac{f(b) - f(a)}{b - a}
\]

Here, \( a = c \), \( b = 0 \), and the average rate of change is given as \(-2\). Applying the formula:

\[
-2 = \frac{f(0) - f(c)}{0 - c}
\]

### Step 1: Find \( f(0) \)
Substitute \( x = 0 \) into the function:

\[
f(0) = \frac{4(0)^2 - 5(0)}{3} = 0
\]

### Step 2: Set up the equation
Now, substituting \( f(0) = 0 \) into the average rate of change formula:

\[
-2 = \frac{0 - f(c)}{0 - c} = \frac{-f(c)}{-c} = \frac{f(c)}{c}
\]

So we have the equation:

\[
-2 = \frac{f(c)}{c}
\]

### Step 3: Find \( f(c) \)
Now, substitute \( f(c) = \frac{4c^2 - 5c}{3} \) into the equation:

\[
-2 = \frac{\frac{4c^2 - 5c}{3}}{c}
\]

Simplify the right-hand side:

\[
-2 = \frac{4c^2 - 5c}{3c}
\]

This simplifies further to:

\[
-2 = \frac{4c - 5}{3}
\]

### Step 4: Solve for \( c \)
Multiply both sides of the equation by 3 to eliminate the denominator:

\[
-6 = 4c - 5
\]

Now, solve for \( c \):

\[
4c = -6 + 5 = -1
\]

\[
c = \frac{-1}{4}
\]

Thus, the value of \( c \) is \( \boxed{-\frac{1}{4}} \).

Would you like more details or have any questions?

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Here are 5 related questions to expand on the concept:
1. What is the significance of the average rate of change in calculus?
2. How does the average rate of change relate to the derivative of a function?
3. Can the average rate of change be positive over some intervals and negative over others?
4. What would change if the average rate of change was given as a different value?
5. How does the concept of average rate of change apply to real-world problems?

**Tip:** The average rate of change between two points is equivalent to the slope of the secant line between those points.

Solve the problem of finding the value of c such that the average rate of change of the function f(x) = (4x^2 - 5x) / 3 is -2 over the interval [c, 0]. This detailed solution explores key algebraic concepts like quadratic functions and the formula for average rate of change. Suitable for Grades 9-12.

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