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

Question

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. — Wuestenfux, Aug 26 '20 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 — Aditya Dwivedi, Aug 26 '20 at 11:49
In your example $(0;;0)$ is an inflexional point with 'horizontal' tangent — Raffaele, Aug 26 '20 at 11:56
@Aditya if a function has extrema than will double derivative is sufficient to get maxima/minima — SS Varshini, Aug 26 '20 at 11:57
I don't think that you are right but most of the times it works — Aditya Dwivedi, Aug 26 '20 at 12:00
@Aditya what if it won't work, Does it implies the function doesn't have maxima/minima — SS Varshini, Aug 26 '20 at 12:04
No then you have do what I said in the previous comment if you want to check local Maxima or minima and the method works all the time — Aditya Dwivedi, Aug 26 '20 at 12:35

score 0 · Answer 1 · answered Aug 26 '20 at 11:51

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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 $- \infty$.

answered Aug 26 '20 at 11:51

Bill O'Haran

if I have an interval of values for x, I can get maximum or minimum for that polynomial $f(x)=x^5+3$ – SS Varshini Aug 26 '20 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. – Bill O'Haran Aug 26 '20 at 11:59

TheSilverDoe · Answer 2 · 2020-08-26T11:58:20.730

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

edited Aug 26 '20 at 11:58

answered Aug 26 '20 at 11:53

TheSilverDoe

I think the op is talking about local extrema not absolute – Aditya Dwivedi Aug 26 '20 at 11:54
Ok I edited my post – TheSilverDoe Aug 26 '20 at 11:58

2 Answers2