HP (Hewlett-Packard) 50g ユーザーズマニュアル

Page 13-12

The interpretation of the variation table shown above is as follows: the function 
F(X) increases for X in the interval (-

∞, -1), reaching a maximum equal to 36 at 

X = -1.  Then, F(X) decreases until X = 11/3, reaching a minimum of –400/27.  
After that F(X) increases until reaching +

∞.   Also, at X = ±∞, F(X) = ±∞.

“Extreme points,” or extrema, is the general designation for maximum and 
minimum values of a function in a given interval.  Since the derivative of a 
function at a given point represents the slope of a line tangent to the curve at 
that point, then values of x for which f’(x) =0 represent points where the graph 
of the function reaches a maximum or minimum.   Furthermore, the value of the 
second derivative of the function, f”(x), at those points determines whether the 
point is a relative or local maximum [f”(x)<0] or minimum [f”(x)>0].  These 
ideas are illustrated in the figure below.

In this figure we limit ourselves to determining extreme points of the function y = 
f(x) in the x-interval [a,b].  Within this interval we find two points, x = x

m

 and x 

= x

M

, where f’(x)=0.  The point x = x

m

, where f”(x)>0, represents a local 

minimum, while the point x = x

M

, where f”(x)<0, represents a local maximum.  

From the graph of y = f(x) it follows that the absolute maximum in the interval 
[a,b] occurs at x = a, while the absolute minimum occurs at x = b.  

For example, to determine where the critical points of function 'X^3-4*X^2-
11*X+30' occur, we can use the following entries in ALG mode: