Statement 1: If $27 a + 9 b + 3 c + d = 0,$ then the equation $f (x) = 4 a x^{3} + 3 b x^{2} + 2 c x + d = 0$ has at least one real root lying in interval $(0, 3)$
Statement 2: If $f (x)$ is continuous in $[a, b],$ derivable in $(a, b)$ such that $f (a) = f (b),$ then there exist at least one point $c \in (a, b)$ such that $f' (c) = 0$

see full answer

Your Exam Success, Personally Taken Care Of

1:1 expert mentors customize learning to your strength and weaknesses – so you score higher in school , IIT JEE and NEET entrance exams.

An Intiative by Sri Chaitanya

Statement-1 is true, statement-2 is true and statement-2 is correct explanation for statement-1.

Statement-1 is true, statement-2 is true and statement-2 is NOT the correct explanation for statement-1.

Statement-1 is true, statement-2 is false.

Statement-1 is false, statement-2 is true.

answer is A.

(Unlock A.I Detailed Solution for FREE)

Best Courses for You

JEE

NEET

Foundation JEE

Foundation NEET

CBSE

Detailed Solution

Consider $f' (x) = 4 a x^{2} + 3 b x^{2} + 2 c x + d$
$\Rightarrow f (x) = a x^{4} + b x^{2} + c x^{2} + d x + e$
$f (0) = e a n d f (3) = 81 a + 27 b + 9 c + 3 d + e$
$= 3 (27 a + 9 b + 3 c + d) + e = e$
Hence, Rolle’s theorem is applicable for $f (x)$
$\Rightarrow$ there exists at least one c in $(a, b)$ such that $f' (c) = 0$