Math Problem Statement

Remember this statememnt can possibly be answered as "either true or false"

There exists a value c in the open interval -2 < x < 0 such that f'(c) = 14/3.

Solution

To determine the validity of the statement, we can apply the Mean Value Theorem (MVT):


Mean Value Theorem:

If a function ff is continuous on a closed interval [a,b][a, b] and differentiable on the open interval (a,b)(a, b), then there exists at least one c(a,b)c \in (a, b) such that:

f(c)=f(b)f(a)ba.f'(c) = \frac{f(b) - f(a)}{b - a}.


Given Data:

  • The interval is [2,0][-2, 0], where ff is continuous and differentiable.
  • Function values:
    • f(2)=5f(-2) = 5
    • f(1)=10f(-1) = 10
    • f(0)=19f(0) = 19
  • The statement asks if there exists cc such that f(c)=143f'(c) = \frac{14}{3}.

Step 1: Compute the Average Rate of Change

To check if f(c)=143f'(c) = \frac{14}{3} is achievable, calculate the average rate of change of ff on the entire interval [2,0][-2, 0]:

Average rate of change=f(0)f(2)0(2)=1952=142=7.\text{Average rate of change} = \frac{f(0) - f(-2)}{0 - (-2)} = \frac{19 - 5}{2} = \frac{14}{2} = 7.

Thus, the average rate of change over the interval [2,0][-2, 0] is 7.


Step 2: Compare 143\frac{14}{3} to 7

  • 1434.67\frac{14}{3} \approx 4.67, which is less than 7.

By the Mean Value Theorem, there exists at least one point cc in the interval (2,0)(-2, 0) where f(c)=7f'(c) = 7, the average rate of change. However, the statement specifically asks for f(c)=143f'(c) = \frac{14}{3}, which is not equal to 7.


Step 3: Analyze the Statement

  • Since the function is differentiable, f(x)f'(x) can take different values within the interval.
  • However, the Mean Value Theorem does not guarantee that f(c)=143f'(c) = \frac{14}{3} because it only ensures the existence of one cc where f(c)f'(c) equals the average rate of change.

Thus, the statement:

"There exists a value cc such that f(c)=143f'(c) = \frac{14}{3}"

cannot be guaranteed.


Final Answer:

The statement must be false.


Do you want me to expand on the Mean Value Theorem or provide more examples? Let me know!


Related Questions:

  1. What is the formal definition of the Mean Value Theorem (MVT)?
  2. How is the average rate of change of a function calculated?
  3. Can f(x)f'(x) take multiple values within an interval?
  4. Under what conditions does the MVT apply?
  5. How is the Mean Value Theorem used to solve real-world problems?

Tip:

The Mean Value Theorem connects the average rate of change of a function to the instantaneous rate of change at some point, making it a fundamental result in calculus.

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Math Problem Analysis

Mathematical Concepts

Mean Value Theorem
Differentiation
Average Rate of Change

Formulas

Mean Value Theorem: f'(c) = (f(b) - f(a)) / (b - a)
Average Rate of Change: (f(b) - f(a)) / (b - a)

Theorems

Mean Value Theorem

Suitable Grade Level

Grades 11-12 (Calculus)