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Limits and Derivatives — Maths STD 11 Science — Question

CBSE BoardEnglish MediumSTD 11 ScienceMathsLimits and Derivatives1 Mark
MCQ
Let $f(x) = (x - a) (x - b) (x - c), a < b < c.$ Then $f(x) = 0$ has two roots. At which interval does these roots belongs?
  • A
    Both the roots in $(a, b)$
  • ✓
    At least one root in $(a, b)$ and at least one root in $(b, c)$
  • C
    Both the roots in $(b, c)$
  • D
    Neither in $(a, b)$ nor in $(b, c)$
✓

Answer

Correct option: B.
At least one root in $(a, b)$ and at least one root in $(b, c)$
$f(x)$ being a polynomial is continuous and differentiable for all real values of $x.$
We also have $f(a) = f(b) = f(c).$
If we apply Rolle’s theorem to $f(x)$ in $[a, b]$ and $[b, c]$ we will observe that $f(x) = 0$
will have at least one root in $(a, b)$ and at least one root in $(b, c).$
But $f(x)$ is a polynomial of degree two,
so that $f(x) = 0$
can’t have more than two roots.
It implies that exactly one root of $f(x) = 0$
will lie in $(a, b)$ and exactly one root of $f(x) = 0$ will lie in $(b, c).$
Let $y = f(x)$ be a polynomial function of degree $n.$
If $f(x) = 0$ has real roots only,
then $f(x) = 0, f(x) = 0, … , f^{n-1}(x) = 0$ will have real roots.
It is in fact the general version of above mentioned application,
because if $f(x) = 0$ have all real roots, then between two consecutive roots of $f(x) = 0,$
exactly one root of $f(x) = 0$ will lie.

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