    Question

# Assertion :The equation f(x)(f′′(x))2+f(x)f′(x)f′′′(x)+(f′(x))2f′′(x)=0 has atleast 5 real roots Reason: The equation f(x)=0 has atleast 3 real distinct roots & if f(x)=0 has k real distinct roots, then f′(x)=0 has atleast k-1 distinct roots.

A
Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
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B
Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
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C
Assertion is correct but Reason is incorrect
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D
Both Assertion and Reason are incorrect
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Solution

## The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertionf(x)(f′′(x))2+f(x)f′(x)f′′′(x)+(f′(x))2f′′(x)=0Now, f(x)f′(x)f′′′(x) is the maximum degree part of equation And thus f(x) must have been a six degree polynomialDue to which above equation has atleast five roots.If f(x)=0 has k real rootsf′(x)=0 has (k−1) real rootsf(x)=a0xk+a1xk−1+a2xk−2+.....anf(x)=ka0xk−1+...an−1=0Degree changed from k to k−1Therefore, both assertion and reason are correct and Reason is the correct explanation for assertion.  Suggest Corrections  0    Join BYJU'S Learning Program
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