what is the meaning of year of the graph of the polynomial function of this degree is odd and the leading coefficient is negative
Answers
A polynomial with a positive leading coefficient will always be increasing, as x approaches infinity. Consider a polynomial with a positive leading coefficient and is degree 'n', where 'n' is an integer greater than zero. Differentiating the function tells us whether the function is increasing or decreasing. Since the coefficient of the first term of the derivative is positive (since 'n' and the coefficient is positive), the limit as x approaches infinity is positive infinity, as the highest degree is all that needs to be evaluated. Since the limit is positive, the function increases indefinitely.
Even if you differentiate once more (assuming n > 1), we find that the leading term still has a positive coefficient. This means that the limit as x approaches infinity, is once again, positive. This indicates positive concavity, thus, supporting the argument.
The same argument can be made to claim that polynomials with a negative leading coefficient approach negative infinity, as x approaches infinity.