For a twice differentiable function `f(x),g(x)` is defined as `g(x)=f^(prime)(x)^2+f^(prime)(x)f(x)on[a , e]dot` If for `a

Options

A. 7
B. 4
C. 6
D. 3

Answer

C

Explanation

f(x) is a continuous function such that`f(b)f(c)lt,0,f(c)f(d)lt,0(a)=0 and f(e)=0.` So, f(x) has at least 4 zeros in[a,e]
`:. f'(x)` has minimum 3 zeros in [a,e]
Now,
`g(x)=(f'(x)f(x))`
`rArr g(x)=(d)/(dx)(f'(x))`
`rArr g(x)=(d)/(dx)h(x) ,"where" h(x)=f(x)f'(x)`
Clealry, h(x) has at least 7 zeros in [a,e] Thereforem by Rolle's theorem `(d)/(dx)h` (x) i.e., g(x) has a least six zeros in [a,e]