Let f:(a,b) →R be twice differentiable such that f(x) = ∫ₐˣ g(t)dt for a differentiable function g(x). If f(x) = 0 has exactly five distinct roots in (a, b), then g(x)g'(x) = 0 has at least :
Let f:(a,b) →R be twice differentiable such that f(x) = ∫ₐˣ g(t)dt for a differentiable function g(x). If f(x) = 0 has exactly five distinct roots in (a, b), then g(x)g'(x) = 0 has at least :
Option 1 -
Seven roots in (a, b)
Option 2 -
Twelve roots in (a, b)
Option 3 -
Five roots in (a, b)
Option 4 -
Three roots in (a, b)
Asked by Shiksha User
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1 Answer
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Correct Option - 2
Detailed Solution:f' (x)=g (x)
f" (x)=g' (x)
⇒g (x).g' (x)=f' (x).f" (x)
f (x) has five roots
⇒f' (x) has atleast 4 roots.
And f" (x) has atleast of 3 roots
⇒g (x).g' (x)=0 has atleast 7 roots in (a, b)
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