Question

3. A function $f(t)$ on an interval $I = (a, b)$ has the mean value property if \begin{equation} f\left(\frac{s+t}{2}\right) = \frac{f(s) + f(t)}{2}, \qquad s, t \in I. \end{equation} Show that any affine function $f(t) = At + B$ has the mean value property. Show that any continuous function on $I$ with the mean value property is affine.

          3. A function $f(t)$ on an interval $I = (a, b)$ has the mean value property if
\begin{equation*} f\left(\frac{s+t}{2}\right) = \frac{f(s) + f(t)}{2}, \qquad s, t \in I.
\end{equation*} Show that any affine function $f(t) = At + B$ has the mean value property. Show that any continuous function on $I$ with the mean value property is affine.

3. A function f(t) on an interval I = (a, b) has the mean value property if

f((s+t)/(2)) = (f(s) + f(t))/(2), s, t ∈ I.
Show that any affine function f(t) = At + B has the mean value property. Show that any continuous function on I with the mean value property is affine.

Added by Pamela W.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Sam Sohn

09/19/2020

Step 1

Given an affine function f(t) = At + B, we need to show that it satisfies the mean value property. Show more…

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A function f(t) on an interval I = (a, b) has the mean value property if f((s + t)/2) = (f(s) + f(t))/2, s, t in I. Show that any affine function f(t) = At + B has the mean value property. Show that any continuous function on I with the mean value property is affine. 3. A function f(t) on an interval I = (a, b) has the mean value property if f(s) + f(t) 2 s, t in I. Show that any affine function f(t) = At + B has the mean value property. Show that any continuous function on I with the mean value property is affine.