What is the relation between the roots of f(x) and its derivative

Very often students come across questions where they have to deduce the number of roots of a function based on the number of roots of its derivative and vice versa.  Even though there are exact rules regarding the relation between the roots of a function and its derivatives, students get confused. Here is some of the concepts defining the relation. Let’s take some of the typical scenarios.

In all the cases, assume that the function f(x) is differentiable for all x on the set of real numbers, i.e. R. Naturally; f(x) will be continuous for all x on R too.

  1. If a function f(x) is a polynomial of degree n has p real roots and n-p imaginary roots, what can be said about the number of real roots of its derivative, i.e. f’(x).
  2. If f(x) is a polynomial of degree n has n real roots and no imaginary roots, what can be said about the number of real roots of its derivative, i.e. f’(x).
  3. If f’(x) is a polynomial of degree n has q real roots, what can be said about the nature of roots of f(x).
  4. If f(x) and f’(x) has common roots, how to know about this.

Let’s take these questions one by one.

  1. If the function f(x) of degree n has p real roots, then its derivative has at least p-1 real roots. The keyword here is at least. There are cases when the derivatives can have more real roots than the function. For example, take a simple quadratic equation f(x) = x2 + x + 1 has 0 real roots while its derivative f’(x) = 2x + 1 has one real root. There are other cases such as f(x) = x3 – 3x + 5 and f’(x) = 3x2 – 3. The function f(x) has just one real root while f’(x) has 2 real roots.
  2. If the function f(x) of degree n has n real roots, then f’(x) will have n-1 real roots and f’’(x) will have n-2 real roots etc. The degrees of f’x) and f’’(x) are n-1 and n-2 respectively. This means all the roots are real. This means if all the roots of a function f(x) are real, then all the roots of f’(x), f’’(x), f’’’(x) etc. will be real.
  3. If any of the derivative of f(x), i.e. f’(x), f’’(x), f’’’(x) and more do not have all real roots, then f(x) doesn’t have all real roots. So it is useful test if you have to find if a polynomial has all real roots. Keep differentiating and checking if the derivative has got any complex roots. If yes, then the polynomial doesn’t have all real roots.
  4. If f(x) and f’(x) have common roots, you can take HCF (highest common factor) between f(x) and f’(x). If the HCF is an expression in x, then the roots of that expression are also the common roots of f(x) and f’(x). If the HCF is 1, then there is no common expression between f(x) and f’(x). This means there is no common root.

By the way, do you know how to find HCF of two expressions?

~ Pankaj Priyadarshi

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