# Chapter 2 - Section 2.8 - The Derivative as a Function - 2.8 Exercises: 49

Curve a represents $f$, curve b $f'$ and curve c $f''$.

#### Work Step by Step

First, look at curve c. We see that curve c has 2 local extrema (the point at which the graph changes from going up to down and vice versa). Therefore, if curve represents $f$ or $f'$, one of the other curves must have 2 points at which it crosses the $Ox$ line (to change from positive to negative and vice versa). However, both of the other curves each crosses the $Ox$ line only 1 time. Therefore, curve c must represent $f''$. Now look at curve a and curve b. The only local extrema of curve b is near the $Oy$ line. If curve b represents $f$ and curve a represents $f'$, then at the point of the local extrema of curve b, curve a would pass the $Ox$ line. However, curve a does not pass the $Ox$ line at that point. Therefore, curve a represents $f$ and curve $b$ represents $f'$.

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