Question

Assertion :The equation $f\left(x\right){\left(f^{\prime \prime }\left(x\right)\right)}^2+f\left(x\right)f^{\prime }\left(x\right)f^{\prime \prime \prime }\left(x\right)+{\left(f^{\prime }\left(x\right)\right)}^2{f}^{\prime \prime }\left(x\right)=0$ has atleast 5 real roots Reason: The equation $f\left(x\right)=0$ has atleast 3 real distinct roots & if $f\left(x\right)=0$ has k real distinct roots, then $f^{\prime }\left(x\right)=0$ has atleast k-1 distinct roots.

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

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

$f\left(x\right){\left(f^{\prime \prime }\left(x\right)\right)}^2+f\left(x\right)f^{\prime }\left(x\right)f^{\prime \prime \prime }\left(x\right)+{\left(f^{\prime }\left(x\right)\right)}^2{f}^{\prime \prime }\left(x\right)=0$

Now, $f\left(x\right)f^{\prime }\left(x\right)f^{\prime \prime \prime }\left(x\right)$ is the maximum degree part of equation

And thus $f\left(x\right)$ must have been a six degree polynomial

Due to which above equation has atleast five roots.

If $f\left(x\right)=0$ has k real roots

$f^{\prime }\left(x\right)=0$ has $(k-1)$ real roots

$f\left(x\right)=a_0{x}^k+a_1{x}^{k-1}+a_2{x}^{k-2}+.....a_n$

$f\left(x\right)=k{a}_0{x}^{k-1}+...a_{n-1}=0$

Degree changed from k to $k-1$

Therefore, both assertion and reason are correct and Reason is the correct explanation for assertion.