The table gives selected values of the differentiable function $f$ . Below is Leo's attempt to write a formal justification for the fact that there exists a value $c$ where $a < c < b$ such that $f'(c) = \frac{f(b) - f(a)}{b - a}$ . Is Leo's justification complete? If not, why? Leo's justification: $f$ is differentiable on $[a, b]$ . So, according to the mean value theorem, there exists a value $c$ somewhere between $a$ and $b$ such that $f'(c) = \frac{f(b) - f(a)}{b - a}$ .

Full solution

Q. The table gives selected values of the differentiable function

f

. Below is Leo's attempt to write a formal justification for the fact that there exists a value

c

where

a < c < b

such that

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

. Is Leo's justification complete? If not, why? Leo's justification:

f

is differentiable on

[a, b]

. So, according to the mean value theorem, there exists a value

c

somewhere between

a

and

b

such that

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

Check Criteria: To apply the Mean Value Theorem, we need to ensure that the function $f$ is continuous on the closed interval $[a, b]$ and differentiable on the open interval $(a, b)$ . We also need the values of $f$ at $a$ and $b$ to calculate the average rate of change.
Calculate Average Rate: Assuming the function meets the criteria for the Mean Value Theorem, we calculate the average rate of change of $f$ on $[a, b]$ by using the formula $\frac{f(b) - f(a)}{b - a}$ .
State Mean Value Theorem: Next, we would state the Mean Value Theorem, which says that if $f$ is continuous on $[a, b]$ and differentiable on $(a, b)$ , then there exists at least one $c$ in $(a, b)$ such that $f'(c)$ is equal to the average rate of change of $f$ on $[a, b]$ .
Verify Leo's Justification: We would then check Leo's justification to see if he has correctly identified the function values at $a$ and $b$ , calculated the average rate of change, and correctly stated the conditions under which the Mean Value Theorem applies.
Evaluate Completeness: If Leo's justification includes all the necessary components and calculations, then it is complete. If any information is missing or incorrect, then the justification is not complete.