Calculus & Analytic Geometry I, MTH 168 

(Instructor: M. USMAN) 

Lab#4 

Objectives: In this lab we will learn to find the critical numbers and absolute maximum and minimum values of a function. 

> restart;with(student):
 

Critical Points Using Solve 

solve(equations, variables) 

 

Let us see few examples of finding the critical points and plotting tangents at critical points 

 

> f:=x->2*x^3-3*x^2-5*x+6;
 

proc (x) options operator, arrow; `+`(`*`(2, `*`(`^`(x, 3))), `-`(`*`(3, `*`(`^`(x, 2)))), `-`(`*`(5, `*`(x))), 6) end proc (1.1)
 

> derf:=diff(f(x),x);
 

`+`(`*`(6, `*`(`^`(x, 2))), `-`(`*`(6, `*`(x))), `-`(5)) (1.2)
 

> points:=solve(derf=0.0,x);
 

1.540833000, -.5408329997 (1.3)
 

> showtangent(f(x), points[2],x=-6..6,y=-10..10);
 

Plot_2d
 

> g:=t->t^3+3*t^2-24*t;
 

proc (t) options operator, arrow; `+`(`*`(`^`(t, 3)), `*`(3, `*`(`^`(t, 2))), `-`(`*`(24, `*`(t)))) end proc (1.4)
 

> derg:=diff(g(t),t);
 

`+`(`*`(3, `*`(`^`(t, 2))), `*`(6, `*`(t)), `-`(24)) (1.5)
 

> cpnts:=solve(derg=0.0,t);
 

2., -4. (1.6)
 

> showtangent(g(t), points[2]);
 

Plot_2d
 

Example: Find the critical points of the function Typesetting:-mrow(Typesetting:-mi(and use a graph to estimate the minimum value of Typesetting:-mrow(Typesetting:-mi(.  (Recall that critical points are the x-values where the derivative is zero.) 

> f:=x->3*x^4+4*x^3-6*x^2;
 

proc (x) options operator, arrow; `+`(`*`(3, `*`(`^`(x, 4))), `*`(4, `*`(`^`(x, 3))), `-`(`*`(6, `*`(`^`(x, 2))))) end proc (1.7)
 

Step 1: Use Maple's differentiation abilities to find the derivative of Typesetting:-mrow(Typesetting:-mi(. 

 

> dervif:=diff(f(x),x);
 

`+`(`*`(12, `*`(`^`(x, 3))), `*`(12, `*`(`^`(x, 2))), `-`(`*`(12, `*`(x)))) (1.8)
 

 

Step 2: Use Maple's algebra ability to determine what value(s) of Typesetting:-mrow(Typesetting:-mi(make this derivative equal zero. 

 

> solve(dervif=0,x);
 

0, `+`(`-`(`/`(1, 2)), `*`(`/`(1, 2), `*`(`^`(5, `/`(1, 2))))), `+`(`-`(`/`(1, 2)), `-`(`*`(`/`(1, 2), `*`(`^`(5, `/`(1, 2)))))) (1.9)
 

 

Step 3: Use the evalf command to estimate the size of the numbers you found as critical point.  

 

> evalf(%);
 

0., .6180339880, -1.618033988 (1.10)
 

 

Step 4: Plot the function Typesetting:-mrow(Typesetting:-mi( on an interval large enough to show the behavior of Typesetting:-mrow(Typesetting:-mi(at both of these critical points. 

 

> plot(f(x),x=-2..2);
 

Plot_2d
 

Absolute Maximum and Minimum 

To find the absolute maximum and minimum of a continuous function on a closed interval, we must take its extreme values at either its critical numbers in the interval or the endpoints of the interval.  For example, on Typesetting:-mrow(Typesetting:-mfenced(Typesetting:-mrow(Typesetting:-mo(, consider the function 

> restart;
 

> f:=x->2*x^3-3*x^2-12*x+1;
 

proc (x) options operator, arrow; `+`(`*`(2, `*`(`^`(x, 3))), `-`(`*`(3, `*`(`^`(x, 2)))), `-`(`*`(12, `*`(x))), 1) end proc (1)
 

To find the critical numbers of Typesetting:-mrow(Typesetting:-mi( in Typesetting:-mrow(Typesetting:-mfenced(Typesetting:-mrow(Typesetting:-mo(, we must compute the derivative of Typesetting:-mrow(Typesetting:-mi( (call it Typesetting:-mrow(Typesetting:-mi(): 

> fprime:=D(f);
 

proc (x) options operator, arrow; `+`(`*`(6, `*`(`^`(x, 2))), `-`(`*`(6, `*`(x))), `-`(12)) end proc
 

> plot({f,fprime},-2..3);
 

Plot_2d
 

> solve(fprime(x)=0,x);
 

2, -1 (2)
 

 

Typesetting:-mrow(Typesetting:-mi( has a root at -1 and 2 

 

Next, compute the value of Typesetting:-mrow(Typesetting:-mi( at the critical numbers and the endpoints: 

> f(-2),f(-1),f(2),f(3);
 

-3, 8, -19, -8
 

We see that the absolute maximum occurs at Typesetting:-mrow(Typesetting:-mi(, y = f(Typesetting:-mrow(Typesetting:-mo()=8 and the absolute minimum at Typesetting:-mrow(Typesetting:-mi(, y = f(2)=-19. 

 

 

Homework Problems:  

1) Find the critical numbers of f(θ)=2cos(θ)+(sin(θ))^2. 

2) Find the critical numbers of f(x)=Typesetting:-mrow(Typesetting:-mi( 

3) Find the critical numbers of  f(t)=t sqrt(`+`(4, `-`(`*`(`^`(t, 2))))) 

4) Find the absolute maximum and absolute minimum values of  f(x)= x^3-6x^2+9x+2Typesetting:-mrow(Typesetting:-mo( 

Error, unable to parse
 

Typesetting:-mrow(Typesetting:-mspace(height =