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The following figure shows the graph of a differentiable function $\mathrm{y}=\mathrm{f}(\mathrm{x})$ on the interval $[\mathrm{a}, \mathrm{b}]$ (not containing 0 ).
Let $g(x)=f(x) / x$ which of the following is a possible graph of $y=g(x) ?$
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Let $g(x)=f(x) / x$ which of the following is a possible graph of $y=g(x) ?$
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Verified Answer
The correct answer is:
Fig. 2
$$
\begin{array}{l}
f^{\prime}(c)=0 \quad f^{\prime}\left(c^{-}\right)>0 \quad f^{\prime}\left(c^{+}\right) < 0 \\
g(x)=\frac{f(x)}{x} \quad g^{\prime}(x)=\frac{x f^{\prime}(x)-1}{x^{2}} \\
g^{\prime}\left(c^{+}\right)=\lim _{h \rightarrow 0} \frac{(c+h) f^{\prime}(c+h)-1}{(c+h)^{2}} < 0 \quad \quad\left(f^{\prime}(c+h) < 0\right)
\end{array}
$$
hence fig.(2)
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