Home
Class 12
MATHS
Let f(x)=x^(3)+ax^(2)+bx+c be the given ...

Let `f(x)=x^(3)+ax^(2)+bx+c` be the given cubic polynomial and `f(x)=0` be the corresponding cubic equation, where `a, b, c in R.` Now, `f'(x)=3x^(2)+2ax+b`
Let `D=4a^(2)-12b=4(a^(2)-3b)` be the discriminant of the equation `f'(x)=0`.
If `D=4(a^(2)-3b)gt0 and f(x_(1)).f(x_(2)) lt0" where " x_(1),x_(2)` are the roots of f(x), then
a. f(x) has all real and distinct roots
b. f(x) has three real roots but one of the roots would be repeated
c. f(x) would have just one real root
d. None of the above

A

f(x) has all real and distinct roots

B

f(x) has three real roots but one of the roots would be repeated

C

f(x) would have just one real root

D

None of the above

Text Solution

Verified by Experts

The correct Answer is:
C
Promotional Banner

Topper's Solved these Questions

  • DY / DX AS A RATE MEASURER AND TANGENTS, NORMALS

    ARIHANT MATHS|Exercise EXAMPLE|6 Videos

  • DY / DX AS A RATE MEASURER AND TANGENTS, NORMALS

    ARIHANT MATHS|Exercise SINGLE OPTION CORRECT TYPE QUESTIONS|10 Videos

  • DIFFERENTIATION

    ARIHANT MATHS|Exercise Exercise For Session 10|4 Videos

  • ELLIPSE

    ARIHANT MATHS|Exercise Exercise (Questions Asked In Previous 13 Years Exam)|26 Videos

Similar Questions

Explore conceptually related problems

Let f(x)=x^(3)+ax^(2)+bx+c be the given cubic polynomial and f(x)=0 be the corresponding cubic equation, where a, b, c in R. Now, f'(x)=3x^(2)+2ax+b Let D=4a^(2)-12b=4(a^(2)-3b) be the discriminant of the equation f'(x)=0 . If D=4(a^(2)-3b)gt0 and f(x_(1)).f(x_(2))=0" where " x_(1),x_(2) are the roots of f(x) , then

Let f(x)=x^(3)+ax^(2)+bx+c be the given cubic polynomial and f(x)=0 be the corresponding cubic equation, where a, b, c in R. Now, f'(x)=3x^(2)+2ax+b Let D=4a^(2)-12b=4(a^(2)-3b) be the discriminant of the equation f'(x)=0 . IF D=4(a^(2)-3b)=0 . Then,

Let f(x)=x^(3)+ax^(2)+bx+c be the given cubic polynomial and f(x)=0 be the corresponding cubic equation, where a, b, c in R. Now, f'(x)=3x^(2)+2ax+b Let D=4a^(2)-12b=4(a^(2)-3b) be the discriminant of the equation f'(x)=0 . IF D=4(a^(2)-3b)lt0 . Then, a. f(x) has all real roots b. f(x) has one real and two imaginary roots c. f(x) has repeated roots d. None of the above

Let f(x)=int_1^xsqrt(2-t^2)dt Then the real roots of the equation x^2-f^(prime)(x)=0 are

A polynomial of 6th degree f(x) satisfies f(x)=f(2-x), AA x in R , if f(x)=0 has 4 distinct and 2 equal roots, then sum of the roots of f(x)=0 is

Let a in R and f : R rarr R be given by f(x)=x^(5)-5x+a , then (a) f(x)=0 has three real roots if a gt 4 (b) f(x)=0 has only one real root if a gt 4 (c) f(x)=0 has three real roots if a lt -4 (d) f(x)=0 has three real roots if -4 lt a lt 4

Let f(x) be a polynomial of degree 5 such that f(|x|)=0 has 8 real distinct , Then number of real roots of f(x)=0 is ________.

Consider the cubic equation f(x)=x^(3)-nx+1=0 where n ge3, n in N then f(x)=0 has

If a,b,c are real roots of the cubic equation f(x)=0 such that (x-1)^(2) is a factor of f(x)+2 and (x+1)^(2) is a factor of f(x)-2, then abs(ab+bc+ca) is equal to

If for all real values of a one root of the equation x^2-3ax + f(a)=0 is double of the other, then f(x) is equal to