**• **At the beginning the function is F: x → x^4+5*x^3-20*x+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^4+5*x^3-20*x+2, ie pow(x,4)+5*pow(x,3)-20*x+2 ] y = pow(x,4)+5*pow(x,3)-20*x+2 return y }}

Maximum between -3 and 0:

Minimum between 0 and 2:

For graphs see** here**.

**• **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 }}

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 _{ x → 0 }F(x)

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.