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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?

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  • $\begingroup$ Have you considered the graph of the function? This will help a lot. $\endgroup$
    – Wuestenfux
    Aug 26, 2020 at 11:48
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    $\begingroup$ 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 $\endgroup$ Aug 26, 2020 at 11:49
  • $\begingroup$ In this case the function doesn't have any extrema at 0 $\endgroup$ Aug 26, 2020 at 11:50
  • $\begingroup$ can we have a function without extrema other than constant $\endgroup$ Aug 26, 2020 at 11:51
  • $\begingroup$ @Ss yes there are many functions $\endgroup$ Aug 26, 2020 at 11:53

2 Answers 2

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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 $- \infty$.

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  • $\begingroup$ if I have an interval of values for x, I can get maximum or minimum for that polynomial $f(x)=x^5+3$ $\endgroup$ Aug 26, 2020 at 11:55
  • $\begingroup$ 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. $\endgroup$ Aug 26, 2020 at 11:59
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Hint : The polynomial is strictly increasing on $\mathbb{R}$, so it has no extremum.

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

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