Most Important Selected Qs for JEE AdvancedMathematicsApplication of Derivatives

Let f(x) be a derivable function and f( )=f( )=0(a < ) , then in the interval ( , ) :

Options

Af(x)+f^ (x)=0 has at least one real root
Bf(x)-f^ (x)=0 has at least one real root
Cf(x) f^ (x)=0 has at least one real root
D(f^ (x) )^2+f(x) f^ (x)=0 has at least two real roots

Correct answer

D. (f^ (x) )^2+f(x) f^ (x)=0 has at least two real roots

Step-by-step solution

(a) Let g(x)=e^x f(x) g( )=g( )=0 According to Rolle's theorem, g^ (x)=e^x f(x)+e^x f^ (x)=0 f(x)+f^ (x)=0 will have at least one real root. (b) Let h(x)=e^ -x f(x) similarly from Rolle's theorem f(x)-f^ (x) will have at least one real root. (c) and (d) f^ (x)=0 has at least one real root ( , ) f(x) f^ (x)=0 will have roots as , , Its derivative (f^ (x) )^2+f(x) f^ (x)=0 will have at least two real roots.

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