Local extrema

Japanese school year: Math II

What you learn

This topic explores finding local maxima and minima of functions by constructing sign charts of first derivatives. It is vital for graphing functions accurately and solving real-world optimization problems such as maximizing efficiency or minimizing production costs. Prerequisites include differentiating polynomials and solving algebraic inequalities to find critical points.

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Key points

This condition is used to find candidates for local peaks and valleys (extrema). When a smooth curve reaches an extremum at $x = a$ , the tangent slope $f'(a)$ is always 0 (horizontal).

f'(a) = 0

A slope of 0 ( $f'(a) = 0$ ) does not always guarantee an extremum. To confirm a true extremum, verify whether the derivative $f'(x)$ changes sign across that point, from positive to negative or from negative to positive.

The endpoints of a given interval are not considered local extrema because they cannot be compared with points on both sides. Local extrema (local peaks and valleys) are distinct from the absolute maximum or minimum values across the entire interval.

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Practice 1Coming soon