Let f ( x) = α( x)β( x) γ ( x) for all real x, where α(x), β(x) and γ ( x) are differentiable functions of x. If f ' (2) = 18 f (2),α' (2) = 3α(2), β' (2) = -4β(2) and γ'(2) = kγ (2) , then the value of k is

Question

Let f ( x) = α( x)β( x) &gamma; ( x) for all real x, where α(x), β(x) and &gamma; ( x) are differentiable functions of x. If f ' (2) = 18 f (2),α' (2) = 3α(2), β' (2) = -4β(2) and &gamma;'(2) = k&gamma; (2) , then the value of k is (A) 14 (B) 16 (C) 19 (D) None of these

Tardigrade Admin · Accepted Answer

We have, f ( x) = α( x)β( x) &gamma; ( x) , for all real x ⇒ f' ( x) = α'( x)β( x) &gamma; ( x) + α( x)β'( x) &gamma; ( x) +α( x)β( x) &gamma;'( x) ⇒ f' (2) = α' (2)β(2) &gamma; (2) + α(2)β'(2)g (2)+a( 2)β(2) &gamma;'(2) 18 f (2) = 3α( 2)β(2) &gamma; (2) - 4α(2)β( 2) &gamma; (2)+ k α(2)β( 2) &gamma; (2) [∵ f'( 2) =18 f (2),α'( 2) = 3α (2) ,β'(2)= -4β(2) and &gamma;'(2) = k&gamma; (2)] ⇒ 18 f (2) = (-1+ k )α( 2)β( 2) &gamma; (2) = (k -1) f ( 2) [∵ f (2) = α(2)β(2) &gamma; (2)] ⇒ k -1 = 18 ∴ k = 19

Q. Let f(x)=α(x)β(x)γ(x) for all real x, where α(x),β(x) and γ(x) are differentiable functions of x. If f′(2)=18f(2),α′(2)=3α(2),β′(2)=−4β(2) and γ′(2)=kγ(2) , then the value of k is

14

16

19

None of these

Q. Let $f (x) = α (x) β (x) γ (x)$ for all real x, where $α (x), β (x)$ and $γ (x)$ are differentiable functions of x. If $f^{'} (2) = 18 f (2), α^{'} (2) = 3 α (2), β^{'} (2) = - 4 β (2)$ and $γ^{'} (2) = kγ (2)$ , then the value of k is