Consider the function $f(x)=x^5-80x+c$

Question

Could someone please help me with the following question:

Consider the function $f(x)=x^5-80x+c$, for an arbitrary constant $c \in R$

(a) Being careful to state and verify hypotheses for any theorem used, show that $f(x)=0$ has at most three solutions.

(b) Establish rigorously how many real solutions $x^5-80x+128=0$ has, using part (a) (hint: look for local extrema for $g(x)=x^5-80x+128$.

Thanks.

Answer 1

(a)

Let us differentiate f(x), we get

$f^{’}(x) = 5x^{4} -80$ $= 5(x^{4}-16)$ $= 5(x^{2}+4)(x + 2)(x - 2)$

You have two real roots. Plug them into the original function. You can find local maxima and minima, changing constant c enable the curve move up and down. You can check how many solutions for $f(x) = 0$. If you can find local maxima and local minima have the opposite sign, you have the three solutions for $f(x) = 0$

Same strategy for (b)