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Statement-1 : The equation 3x^(5)+15x-18...

Statement-1 : The equation `3x^(5)+15x-18=0` has exactly one real root.
Statement-2: Between any two roots of , there is a root of its derivative f'(x) .

A

Statement-1 is True, Statement-2 is Ture, Statement-2 is a correct explanation for statement-1

B

Statement-1 is True, Statement-2 is Ture, Statement-2 is not a correct explanation for statement-1

C

Statement-1 is True, Statement-2 is False

D

Statement-1 is False, Statement-2 True.

Text Solution

Verified by Experts

The correct Answer is:
A
Statement-, being algebraic intepretition of Rolle's theorem, is true,
Let `x_(1),x_(2)` be two roots of `f(x)=3x^(5)+15x-18`. Then `f'(x)=12x^(4)+15` must have a root between `x_(1) and x_(2)` . But, `f(x)le0` for all x, So, it has no real root. This contradicts the algebraic interpertaion of Rolle's theorem.
Hence `f(x)=3x^(5)+15x-18` has exactly one real root x=1
Thus, both the satement are true and statement-2 is correct explanation for statement-1
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Knowledge Check

  • Statement-1: The equation e^(x-1) +x-2=0 has only one real root. Statement-2 : Between any two root of an equation f(x)=0 there is a root of its derivative f'(x)=0

    A
    Statement-1 is True, Statement-2 is Ture, Statement-2 is a correct explanation for statement-1
    B
    Statement-1 is True, Statement-2 is Ture, Statement-2 is not a correct explanation for statement-1
    C
    Statement-1 is True, Statement-2 is False
    D
    Statement-1 is False, Statement-2 True.

  • Statement-1: There is a value of k for which the equation x^(3) - 3x + k = 0 has a root between 0 and 1. Statement-2: Between any two real roots of a polynomial there is a root of its derivation.

    A
    Statement-1 is True, Statement-2 is True, Statement-2 is a correct explanation for Statement-1.
    B
    Statement-1 is True, Statement-2 is True, Statement-2 is not a correct explanation for Statement-1.
    C
    Statement-1 is True, Statement-2 is False.
    D
    Statement-1 is False, Statement-2 is True.

  • If the equation x^(2)-kx+1=0 has no real roots then

    A
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    B
    `kgt2`
    C
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    D
    None of these

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