You visited us 1 times! Enjoying our articles? Unlock Full Access!

Byju's Answer

Standard XII

Mathematics

Nature of Roots

18. Let f(x) ...

Question

18. Let f(x) = ax + bx + cx + d be a cubic polynomial (a, b, c, d R). If f(m) f(n) = 0 where m and n are the distinct real roots of f'(x) = 0, then (A) f(x) = 0 has all three different real roots (B) f(x) = 0 has three real roots but two of them are equal (C) f(x) = 0 has only one real root (D) all three roots of f(x) = 0 are real and equal

Open in App

Solution

Suggest Corrections

Similar questions

Q. Statement 1 : If

f (x) = a x^{2} + b x + c

, where

a > 0, c < 0

and

b \in R

, then roots of

f (x) = 0

must be real and distinct .
Statement 2 : If

f (x) = a x^{2} + b x + c,

where

a > 0, b \in R, b \neq 0

and the roots of

f (x) = 0

are real and distinct, then

c

is necessarily negative real number .

Q. Assertion :The equation

f (x) {(f^{''} (x))}^{2} + f (x) f^{'} (x) f^{'''} (x) + {(f^{'} (x))}^{2} f^{''} (x) = 0

has atleast 5 real roots Reason: The equation

f (x) = 0

has atleast 3 real distinct roots & if

f (x) = 0

has k real distinct roots, then

f^{'} (x) = 0

has atleast k-1 distinct roots.

Q. Let

f (x) = x^{2} + a x + b,

where a, b

ϵ

R. If

f (x) = 0

has all its roots imaginary, then the roots of

f (x) + f^{'} (x) + f " (x) = 0

are

If f : D $\to$ R $f (x) = \frac{x^{2} + b x + c}{x^{2} + b_{1} x + c_{1}}, w h e r e α, β$ are th roots of the equation $x^{2} + b x + c = 0$ and $α_{1}, β_{1}$ are the roots of $x^{2} + b_{1} x + c_{1} = 0$ . Now, answer the following question for f(x). A combination of graphical and analytical approach may be helpful in solving these problems. If $α_{1} a n d β_{1}$ are real, then f(x) has vertical asymptote at $x = (α_{1}, β_{1}$ ).
If the equations $x^{2}$ + bx + c = 0 and $x^{2} + b_{1} x + c_{1} = 0$ do not have real roots, then

Q. If

f : R \to R

and

f (x)

is a polynomial function of degree eleven and

f (x) = 0

has all real and distinct roots. Then the equation

(f^{'} (x))^{2} - f (x) f^{''} (x) = 0

has

Join BYJU'S Learning Program