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1. relation and function — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMaths1. relation and function1 Mark
MCQ
Let $f(x)$ be a non-constant polynomial with real coefficients such that $f\left(\frac{1}{2}\right)=100$ and $f(x) \leq 100$ for all real $x$. Which of the following statements is NOT necessarily true?
  • A
    The coefficient of the highest degree term in $f(x)$ is negative.
  • B
    $f(x)$ has at least two real roots.
  • ✓
    If $x \neq 1 / 2$ then $f(x) < 100$.
  • D
    At least one of the coefficients of $f(x)$ is bigger than $50.$
✓

Answer

Correct option: C.
If $x \neq 1 / 2$ then $f(x) < 100$.
c
(c)

We have, $f\left(\frac{1}{2}\right)=100$

$f(x) \leq 100, \forall x \in R$

$\therefore \quad f(x)=a\left(x-\frac{1}{2}\right)$

$\left[a_0 x^{n-1}+a_1 x^{n-2}+\ldots+a_{n-1}\right]+100$

If $f(x) \leq 100, \forall x \in R$

$\therefore a < 0$ and $f(x)$ must be even degree polynomial.

Since, there may be more value of $x$ at which $f(x)$ attains maximum.

$\therefore$ If $x \neq \frac{1}{2}$, then $f(x) < 100$ may be false.

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