Roots of equation

Question

i) $x^4+4x+c$ has atmost  _______ roots.

ii)Number of real roots for $x^5 + 10x+3$ ?

Comments

i)  2

ii) 1 ??

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No Ans given

Anyhow please explain your approach !!

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(i) $f(x)=x^{4}+4x+c$

$f'(x)=4x^{3}+4=0$    $\Rightarrow x^{3}=-1$ or $x=-1$

$f''(x=-1)=12> 0$ // this shows that function has only one minima at $x=-1$.

this shows that above function will somewhat look like upper parabola, therfore it can have atmost 2 roots only.

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(ii) $f(x)=x^{5}+10x+3$

$f'(x)=5x^{4}+10>0(always)$

since it is monotonically increasing function, it will cut x axis only once, therefore it will have only 1 root.

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In sol1 u checked minima becoz of c value unknown?

From the observation c cannot be >3 ryt??

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@joshi_nitish q1 asks for number of roots, be it complex or real.

So, at most it can have 4 roots by Fundamental theorem of Algebra.