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Mathematics · Ch 6 — Application of Derivatives

Maxima and Minima -- First Derivative Test

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Maxima and Minima -- First Derivative Test

Local maximum and local minimum, defined. A function ff has a local maximum at x=cx = c if f(c)≥f(x)f(c) \ge f(x) for every xx in some open interval around cc -- the graph has a "peak" there, at least compared to its immediate neighbourhood (it need not be the largest value of ff anywhere on its whole domain). Likewise ff has a local minimum at cc if f(c)≤f(x)f(c) \le f(x) throughout some neighbourhood of cc -- a "valley." Together these are called local extrema, or turning points.

Critical points. A point cc in the domain of ff is a critical point if either f′(c)=0f'(c) = 0 or f′(c)f'(c) does not exist. It is a basic fact (following from the same reasoning as the increasing/decreasing test of Section 2) that every local extremum of a differentiable function occurs at a critical point -- so the search for local maxima and minima always begins by solving f′(x)=0f'(x) = 0.

The geometric idea behind the first derivative test. At a local maximum, the curve is rising just before cc and falling just after -- the tangent slope f′(x)f'(x) is positive to the left of cc and negative to the right, i.e. f′f' changes sign from ++ to −- as xx crosses cc. At a local minimum, the mirror image happens: the curve falls then rises, so f′f' changes from −- to ++. If f′f' does not change sign at a critical point (stays ++ on both sides, or −- on both sides), the curve is simply still rising (or falling) straight through cc -- not a turning point at all.

First Derivative Test. Let ff be continuous at a critical point cc (where f′(c)=0f'(c) = 0), and differentiable in a neighbourhood of cc except possibly at cc itself.

  • If f′(x)>0f'(x) > 0 just to the left of cc and f′(x)<0f'(x) < 0 just to the right, ff has a local maximum at cc.
  • If f′(x)<0f'(x) < 0 just to the left of cc and f′(x)>0f'(x) > 0 just to the right, ff has a local minimum at cc.
  • If f′(x)f'(x) has the same sign on both sides of cc, cc is neither a local maximum nor a local minimum.

Method. Exactly the sign-chart procedure of Section 2, reused with a sharper conclusion: (1) find all critical points by solving f′(x)=0f'(x)=0; (2) test the sign of f′f' in each of the intervals the critical points create; (3) at each critical point, read off the sign change (or its absence) from the two neighbouring intervals to classify it as local max, local min, or neither; (4) substitute the critical xx-value back into ff itself to state the local maximum/minimum value (not just its location) -- see Example 7, where x=−1x=-1 gives local maximum value f(−1)=2f(-1)=2 and x=1x=1 gives local minimum value f(1)=−2f(1)=-2. …

Figure 2A local maximum and a local minimum on a curve

What this figure shows. Shows a smooth wavy curve y = f(x) over an interval of the x-axis, containing one local hump (a point where the curve rises then falls, forming a peak) and one local dip (a point where the curve falls then rises, forming a valley), joined by a falling stretch of curve between them. The peak point is labelled as a local maximum with a horizontal dashed tangent line drawn touching the curve exactly at its top, and the valley point is labelled as a local minimum with a horizontal dashed tangent line touching the curve exactly at its bottom. Small arrows along the curve on either side of each labelled point indicate the direction the curve is rising or falling immediately before and after that point, showing the rising-then-f …

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