# Given a polynomial of degree 5 how to find its maxima or minima

Suppose that $$f(x)= x^{5}+3$$

$$f'(x)=5*x^{4}$$

To get maxima/minima the first-order derivative is equated to $$0$$

$$f'(x)=5*x^{4}=0$$ => $$x=0$$

No matter what the degree of $$x$$, the value of $$x=0$$

How can I get maximum or minima value?

Can we get maxima or minima for any polynomial by second-order derivation?

• Have you considered the graph of the function? This will help a lot. Aug 26, 2020 at 11:48
• You should study the roots of the derivative and check whether its sign changes from negative to positive or the vice verca or is of same sign Aug 26, 2020 at 11:49
• In this case the function doesn't have any extrema at 0 Aug 26, 2020 at 11:50
• can we have a function without extrema other than constant Aug 26, 2020 at 11:51
• @Ss yes there are many functions Aug 26, 2020 at 11:53

This is one of the perfect examples to show that $$f'(x)=0$$ does not always imply that $$x$$ is an extremum.

Look at the graph of your polynomial and you will quickly realise what is happening.

As for the maximum and minimum values of a polynomial with odd degree, remember what happens when $$x$$ tends to either $$+ \infty$$ or $$- \infty$$.

• if I have an interval of values for x, I can get maximum or minimum for that polynomial $f(x)=x^5+3$ Aug 26, 2020 at 11:55
• If you are indeed restraining yourself to some interval $I \subset \mathbb{R}$, you should look at the maximum and minimum of your interval. Again, the easiest way probably is to plot the graph of this polynomial. This should guide your reflexion. Aug 26, 2020 at 11:59

Hint : The polynomial is strictly increasing on $$\mathbb{R}$$, so it has no extremum.

• I think the op is talking about local extrema not absolute Aug 26, 2020 at 11:54
• Ok I edited my post Aug 26, 2020 at 11:58