Question

Cubic polynomial x=f(x)cuts y axis at most

Answer 1

Answer:

A cubic polynomial is a continuous graph which has the form ax^3+bx^2+cx+d. Now when x is a really big number or x tends to infinity ax^3 tends to even bigger number. For some big x ax^3 will be greater than bx^2+cx+d because in comparison to ax^3 the the other lower powers of x are small.( Take an example. Take the number 100. It's square is 10000 buts it's cube is 1000000 which is way bigger than 10000). So we can say for some x the cubic function will be positive. Now when x tends to a very small no or x tends to negative infinity ax^3 tends to even smaller no. In this case ax^3 is way smaller than bx^2+cx+d.( Take an example. Take the number -100. It's square is 10000 but it's cube is -100000 which is way smaller than 10000). So we can say that for some x the cubic will be negative. Since the cubic is a continuous function and goes from a positive value to a negative value or from a negative value to positive value, it has to cross 0 for some x. Hence cubic always crosses 0.

Now let's talk about the sqaured function of the form ax^2+bx+c. When x is a very big no ax^2 will be bigger than bx+c.( 10000 is bigger 100). Hence the function will tend to infinity for x tends to infinity. When x tends to negative infinity ax^2 tends to a very big no too which is bigger than bx+c(10000 is bigger than -100). So for x tending to negative infinity the function still tends to infinity. For all the values of x between infinity and negative infinity let's say there is a choice if they want to be positive or negative or 0. So the squared function may or may not cross 0.