The function \f(x)=x^{3}-6 x^{2}+a x+b is such that \f(2)=\f(4)=0. Consider two statements. (\S_{1}) there exists x_{1}, x_{2} \\in(2,4), x_{1} < x_{2}, such that \f'(x_{1})=-1 and \f'(x_{2})=0. ( (\S_{2}) there exists x_{3}, x_{4} \\in(2,4), x_{3} < x_{4}, such that f is decreasing in (2, x_{4}), increasing in (x_{4}, 4) and 2 \f'(x_{3})=\\sqrt{3} f(x_{4}). Then, [1 Sep 2021 Shift 2]

Correct Answer is : both (S1) and (S2) are true

both (\S_{1}) and (\S_{2}) are true

(\S_{1}) is false and (\S_{2}) is true

both (\S_{1}) and (\S_{2}) are false

(\S_{1}) is true and (\S_{2}) is false

Application of Derivatives Part 2

Section: Mathematics

Question : 69 of 100

Marks: +1, -0

The function f(x)=x3−6x2+ax+b is such that f(2)=f(4)=0. Consider two statements. (S1) there exists x1,x2∈(2,4),x1<x2, such that f′(x1)=−1 and f′(x2)=0. ( (S2) there exists x3,x4∈(2,4),x3<x4, such that f is decreasing in (2,x4), increasing in (x4,4) and 2f′(x3)=√3f(x4). Then,

[1 Sep 2021 Shift 2]

both (S1) and (S2) are true
(S1) is false and (S2) is true
both (S1) and (S2) are false
(S1) is true and (S2) is false

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