16.2.2 Local extrema
The extrema
command attempts to find local extrema of a
univariate/multivariate differentiable expression, possibly subject to
equality constraints, by using analytical methods.
-
extrema takes two mandatory arguments and two
optional arguments:
-
expr, a differentiable expression.
- Optionally, constr, a list of equality constraints,
where each constraint is an equality or an expression assumed to be
equal to zero, and the number of constraints must be strictly less
than the number of variables. If there is only one constraint, it
does not have to be a list.
Additionally, the Jacobian matrix of the constraints must be full
rank (i.e., denoting the kth constraint by
gk(x1,x2,…,xn)=0 for k=1,2,…,m and letting
g=(g1,g2,…,gm), the Jacobian matrix of g
must be equal to m).
- vars, a list of variables. A variable can be
specified with bounds, x=a..b, where a or b is allowed to be
-infinity or infinity. Note that extrema respects
the bound constraints set to variables by the assume command.
If there is only one variable, it does not have to be enclosed in square brackets.
The parameter vars can also be entered as a list of values
of the variables; e.g. [x1=a1,x2=a2,…,xn=an], in which case the critical
point close to a=(a1,a2,…,an) is computed
numerically by applying an iterative method with initial point a.
- Optionally, opt, which can be either
-
order=n, to make n the upper bound for the order of derivatives examined
in the process (so if n=1 only critical points are found, by default n=5), or
- lagrange to specify the method of Lagrange multipliers.
- extrema(expr ⟨,constr ⟩,vars ⟨,opt ⟩)
returns a list [min,max], where min is the list of
local minima and max is the list of local maxima of
expr.
Saddle, unclassified and indeterminate points are reported in the message area.
If lagrange is passed as an optional last argument, the
method of Lagrange multipliers is used. Else, the problem is reduced
to an unconstrained one by applying implicit differentiation.
- If some critical points are left unclassified, you might consider repeating the
process with larger values of the parameter order (e.g. for f(x)=x8,
extrema requires n≥ 8 in order to find the minimum),
although the success is not guaranteed. Indeterminate critical points are
unclassifiable.
Examples
| extrema(-2*cos(x)-cos(x)^2,x) |
| extrema(x/2-2*sin(x/2),x=-12..12) |
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| assume(a>=0):; extrema(x^2+a*x,x) |
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| extrema(exp(x^2-2x)*ln(x)*ln(1-x),x=0.5) |
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| extrema(x^3-2x*y+3y^4,[x,y]) |
[x=0,y=0]: saddle point
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| assume(a>0):; extrema(x/a^2+a*y^2,x+y=a,[x,y],lagrange) |
| extrema(x^2+y^2,x*y=1,[x=0..inf,y=0..inf]) |
To find only the critical points of f(x,y,z)=x y z subject to x+y+z=1:
| extrema(x*y*z,x+y+z=1,[x,y,z],order=1) |
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The Peano surface z=(2x2−y)(y−x2) was proposed by Giuseppe Peano in 1899 as
a counter-example to a criterion for the existence of maxima and minima which was
conjectured at the time. Indeed, it is not possible to classify its only critical
point (0,0), which turns out to be a saddle, not a maximum:
| extrema((2x^2-y)*(y-x^2),[x,y]) |
[x=0,y=0]: indeterminate critical point