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is a local (or relative) maximum of a

) ≥ f(x) for all x with |x-x
| < ε. On a graph of a function, its local maxima will look like the tops of hills.

for which f(x
) ≤ f(x) for all x with |x-x
| < ε. On a graph of a function, its local minima will look like the bottoms of valleys.

for which f(x
) ≥ f(x) for all x. Similarly, a global minimum is a point x
for which f(x
) ≤ f(x) for all x. Any global maximum (minimum) is also a local maximum (minimum); however, a local maximum or minimum need not also be a global maximum or minimum.

| < ε.

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The function $x^2$ has a unique global minimum at x = 0.

The function $x^3$ has no global or local minima or maxima. Although the first derivative (3$x^2$) is 0 at x = 0, the second derivative (6x) is also 0.

The function $x^3$/ 3 - x has first derivative $x^2\; -\; 1$ and second derivative 2x. Setting the first derivative to 0 and solving for x gives stationary points at -1 and +1. From the sign of the second derivative we can see that -1 is a local maximum and +1 is a local minimum. Note that this function has no global maxima or minima.

The function |x| has a global minimum at x = 0 that cannot be found by taking derivatives, because the derivative does not exist at x = 0.

The function cos(x) has infinitely many global maxima at 0, ±2π, ±4π, ..., and infinitely many global minima at ±π, ±3π, ... .

The function 2cos(x) - x has infinitely many local maxima and minima, but no global maxima or minima.

The function $x^3\; +\; 3\; x^2\; -\; 2\; x\; +\; 1$ defined over the closed interval (segment) -4,2 (see graph) has two extrema: one local maximum in $x\; =\; (-1-sqrt)/3$, one local minimum in $x\; =\; (-1+sqrt)/3$, a global maximum on x=2 and a global minimum on x=-4.

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Mechanical equilibrium