4.7 Maxima/Minima Problems

← Chapter 4 §4.7 Maxima/Minima Problems 极值问题 · Chapter 4 · Differentiation of Functions of Several Variables

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Access for free at openstax.org · CC BY-NC-SA 4.0

Objective	Takeaways; say them first, then click
4.7.1 Use partial derivatives to locate critical points.	$f_x=f_y=0$ , or a partial undefined; local vs global; smooth extremum ⇒ critical (Fermat)
4.7.2 Apply a second derivative test.	saddle: critical, no extremum; $D=f_{xx}f_{yy}-f_{xy}^2$ : min, max, saddle, or no conclusion
4.7.3 Examine critical and boundary points for absolute extrema.	closed + bounded ⇒ they exist; compare interior critical points, edges, corners; parametrize a curved edge

Mistake	Result	Fix
$D<0$ read as a maximum	a saddle called a top	$D<0$ : saddle; only $D>0$ gives an extremum
Deciding max/min from $D$ alone	Checkpoint 4.35: $D=68$ is a maximum, not a minimum	look at $f_{xx}$ when $D>0$
$D=0$ read as "no extremum"	Exercise 325: $9-x^4y^4$ has a maximum at the origin	$D=0$ : the test is silent; look at $f$
Skipping points where a partial fails to exist	the tip of $\sqrt{x^2+y^2}+1$ missed (Exercise 317)	list them as critical points
Only interior critical points on a closed region	Example 4.40a: 17 found, the maximum 36 on the edge missed	every edge, and the corners
Reporting the parameter $t$ , not the point	" $t=\tfrac54$ " for the point $(4,\tfrac54)$	substitute back into $x(t),y(t)$

#	task	answer
4.34	critical point of $x^3+2xy-2x-4y$	$(2,-5)$
4.35	local extrema of $x^3+2xy-6x-4y^2$	saddle $(\tfrac43,\tfrac13)$ , $D=-68$ ; local max $(-\tfrac32,-\tfrac38)$ , $D=68$ , value $\tfrac{99}{16}$
4.36	absolute extrema of $4x^2-2xy+6y^2-8x+2y+3$ on $[0,2]\times[-1,3]$	min $f(1,0)=-1$ ; max $f(0,3)=63$

#	function	answer
311	$(3x-2)^2+(y-4)^2$	$(\tfrac23,4)$
313	$15x^3-3xy+15y^3$	$(0,0)$ , $(\tfrac1{15},\tfrac1{15})$
315	$-x^2-5y^2+8x-10y-13$	maximum at $(4,-1,8)$
317	$\sqrt{x^2+y^2}+1$	minimum at $(0,0,1)$
319	$x^2y^2$	test fails; absolute minimum 0 on the axes
321	$2xy+3x+4y$	saddle $f(-2,-\tfrac32)=-6$
323	$x^2+4xy+y^2$	saddle $(0,0,0)$
325	$9-x^4y^4$	$D=0$ ; a maximum, $f(0,0)=9$
327	$3x^2-2xy+y^2-8y$	minimum at $(2,6)$
329	$y^2+xy+3y+2x+3$	saddle $(1,-2)$
331	$x^2+2y^2-x^2y$	saddles $(\pm2,1)$ ; minimum $(0,0)$
333	$e^{-(x^2+y^2+2x)}$	maximum at $(-1,0)$
335	$x^2+10xy+y^2$	saddle $(0,0)$
337	$120x+120y-xy-x^2-y^2$	maximum at $(40,40)$
339	$x^2+x-3xy+y^3-5$	saddle $(\tfrac14,\tfrac12)$ ; minimum $(1,1)$
341	$ye^x-e^y$	saddle $(0,0)$
343	$\sin x\sin y$ on $(0,2\pi)^2$	saddle $(\pi,\pi)$ ; max $(\tfrac\pi2,\tfrac\pi2),(\tfrac{3\pi}2,\tfrac{3\pi}2)$ ; min $(\tfrac\pi2,\tfrac{3\pi}2),(\tfrac{3\pi}2,\tfrac\pi2)$
345	$x^2+y^2-2y+1$ on $x^2+y^2\le4$	min $(0,1,0)$ ; max $(0,-2,9)$
347	$\frac{-2y}{x^2+y^2+1}$ on $x^2+y^2\le4$	min $(0,1,-1)$ ; max $(0,-1,1)$
349	points of $x^2-yz=5$ nearest the origin	$(\pm\sqrt5,0,0)$
351	largest package, length + girth 108 in	18 by 36 by 18 in
353	$z=x^2+y^2+10$ nearest $x+2y-z=0$	$(\tfrac12,1,\tfrac{45}4)$ ; on the plane $(\tfrac{47}{24},\tfrac{47}{12},\tfrac{235}{24})$
355	shoe revenue $R(x,y)$	$x=3$ , $y=6$
357	soda can, height + circumference 120 cm	$V=\tfrac{64{,}000}{\pi}\approx20{,}372\ \text{cm}^3$

Maxima/Minima Problems

Candidates for a max or min: both partials are 0, or one does not exist

Where the partials' denominator is 0, every point is critical: a whole hyperbola

For a polynomial, set both partials to 0 and solve the system: $(-1,-1)$

A local max beats every nearby value; a global max beats every value in the domain

At a smooth local max or min, both partials are 0: the tangent plane is level

Fermat in two variables is Fermat in one variable, used twice

A critical point need not be an extremum: a saddle rises one way and falls another

The discriminant $D$ sorts a critical point: minimum, maximum or saddle

One critical point, $D=144>0$ and $f_{xx}>0$ : a minimum of $-16$

$D>0$ means the surface bends the same way in every direction

When $D=0$ the test is silent: the same zero hides a min, a max or a saddle

One surface, two level spots: the sign of $D$ tells the saddle from the minimum

$D$ can change from point to point: one saddle, one minimum

Drag the point: the tangent plane is level only at a critical point

A saddle at $(\tfrac43,\tfrac13)$ and a local maximum at $(-\tfrac32,-\tfrac38)$

On a closed, bounded region a continuous function always reaches a max and a min

The absolute extremes are at interior critical points or on the boundary: compare them all

On the rectangle the max 36 is at a corner, the min 17 inside

Walk the boundary to find every edge candidate

Maximum profit $648,000: 21,000 balls and 3 hours of advertising a month

At the best plan, one more ball or one more hour gains nothing

Theorem 4.19 is EVT plus Fermat; drop closed or bounded and the max can vanish

Min $-1$ inside at $(1,0)$ , max 63 at the corner $(0,3)$

On a disc, parametrize the circle: max ≈ 44.84 on the edge, min −13 inside

Critical points find the candidates, $D$ sorts them, the boundary finishes the job

Common mistakes

Practice and answers