• At the beginning the function is F: x → x^3−2, that I could study with this program, for polynomial functions.
function F(x) { with(Math) { /// you can change F [ now it is x - > x^3-2, ie pow(x,3)-2 ] return pow(x,3)-2 }}
• To consider another function, we study F: x → x³·cos(x)/( (cos(x)+x)·(sin(x)−x) ) in the interval [-0.5, 5].
function F(x) { with(Math) { y = pow(x,3)*cos(x) / ( (cos(x)+x)*( sin(x)-x ) ) return y }}
(we can draw the graphs with JavaScript; for example, see the script x3cossin - see the code; se also here).
In x=0, where sin(x)-x = 0, it is not defined. It is continuous function in [-1/2,0) ∪ (0,5].
How can we fill the hole in order to obtain a continuous function in [-1/2,5]?
We calculate
function F(x) { with(Math) { y = pow(x,3)*cos(x) / ( (cos(x)+x)*( sin(x)-x ) ) return y }}
We could thicken the tab, but it is "evident" that the limit is 6.
Let's look for the "maximum" of F. With this program (with the new F) I find:
F has the maximum value 4.65229469036 at 3.015673948.