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;

 

(1.1)

 

>	derf:=diff(f(x),x);

 

(1.2)

 

>	points:=solve(derf=0.0,x);

 

(1.3)

 

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

 

>	g:=t->t^3+3*t^2-24*t;

 

(1.4)

 

>	derg:=diff(g(t),t);

 

(1.5)

 

>	cpnts:=solve(derg=0.0,t);

 

(1.6)

 

>	showtangent(g(t), points[2]);

 

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

>	f:=x->3*x^4+4*x^3-6*x^2;

 

(1.7)

 

Step 1: Use Maple's differentiation abilities to find the derivative of . 

>	dervif:=diff(f(x),x);

 

(1.8)

 

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

>	solve(dervif=0,x);

 

(1.9)

 

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

>

evalf(%);

 

(1.10)

 

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

>	plot(f(x),x=-2..2);