Question

If a + b +c = 0, then the equation $3a{x}^2+2bx+c=0$ has, in the interval (0, 1)

• A. Atleast one root
• B. Atmost one root
• C. No root
• D. None of these

Answer 1

The correct option is A

Atleast one root

Let $f\left(x\right)=a_n{X}^n+a_{n-1}{X}^{n-1}+...+a_2{x}^2+a_{1}x+a_0$
Which is a polynomial function in x of degree n. Hence f(x) is continuos and differentiable for all x.
Let $\infty <\beta$ . We given, f $\left(\infty \right)=0=f\left(\beta \right)$.
By Rolle's theorem, f' (c) = 0 for some value c,
$\infty <c<\beta$.
Hence the equation
$F^{\prime }\left(x\right)=n{a}_n{x}^{n-1}+(n-1)a_{n-1}{x}^{n-2}+...+a_1=0$
has atleast one root between $\infty and\beta$.
Hence (c) is the correct answer.