MHB Checking if f(x)=x^3 is Monotonic: No Derivatives Needed

  • Thread starter Thread starter Yankel
  • Start date Start date
  • Tags Tags
    Functions
AI Thread Summary
To determine if the function f(x) = x^3 is monotonic without using derivatives, one can apply the definition of monotonicity. By evaluating the expression f(x+h) - f(x) for a positive real number h, if the result is greater than or equal to zero, the function is monotonically increasing. Conversely, if the result is less than or equal to zero, the function is monotonically decreasing. This approach effectively assesses the function's behavior across intervals. Thus, f(x) = x^3 can be analyzed for monotonicity using this method.
Yankel
Messages
390
Reaction score
0
Hello,

I want to check if

f(x)=x^3 is monotonically increasing or monotonically decreasing or not monotonic at all.

How do I do that, without using derivatives yet ?

Thanks !
 
Mathematics news on Phys.org
You use the definition.
 
Let $h$ be some positive real number, then if:

$$f(x+h)-f(x)\ge0$$ the function is monotically increasing or if:

$$f(x+h)-f(x)\le0$$ the function is monotically decreasing.
 
Seemingly by some mathematical coincidence, a hexagon of sides 2,2,7,7, 11, and 11 can be inscribed in a circle of radius 7. The other day I saw a math problem on line, which they said came from a Polish Olympiad, where you compute the length x of the 3rd side which is the same as the radius, so that the sides of length 2,x, and 11 are inscribed on the arc of a semi-circle. The law of cosines applied twice gives the answer for x of exactly 7, but the arithmetic is so complex that the...
Back
Top