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Q8E

Expert-verifiedFound in: Page 215

Book edition
2nd

Author(s)
James Stewart

Pages
830 pages

ISBN
9781133112280

**To determine the values of \(c\) that satisfies the conclusion of the Mean Value Theorem for the interval \((1,7)\) using the given graph of the function.**

The values of \(c\)that satisfies the conclusion of the Mean Value Theorem for the interval \((1,7)\) by the given graph of the function is \(\underline {c = 0.3,3,6.3} \).

\(f(x)\)is a continuous function on \((1,7)\) and \(f(x)\) is differentiable in \((1,7)\).

**Assume **\(f\)** to be a function satisfying the following properties.**

**(1) **\(f\)**Continuous on the interval **\((a,b)\)**.**

**(2) **\(f\)** Is differentiable on the interval **\((a,b)\)**.**

**Then there is a number **\(c\)** in **\((a,b)\)** such that **\({f^\prime }(c) = \frac{{f(b) - f(a)}}{{b - a}}\)**.**

Here, in the given graph is \(a = 1,\;b = 7\).

Also, from the given graph \(f(x)\) is a continuous function on \(\left( {1,7} \right)\) and \(f(x)\) is differentiable in \((1,7)\).

Clearly, from the graph it can be seen that \(f(1) = 0.5;\;f(7) = 1.5\).

Thus, \({f^\prime }(c)\) is evaluated as shown below.

\(\begin{aligned}{c}{f^\prime }(c) &= \frac{{f(7) - f(1)}}{{7 - 1}}\\ &= \frac{{1.5 - 0.5}}{{7 - 1}}\\ &= \frac{1}{6}\\ &= 0.167\end{aligned}\)

It can be observed from the graph that the values of \(c = 0.3,3,6.3\).

Therefore, the values of \(c\) that satisfies the conclusion of the Mean Value Theorem for the interval \((1,7)\) by the given graph of the function is .

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