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Let F(x)=f(x)g(x)h(x) for all real x ,w ...

Let `F(x)=f(x)g(x)h(x)` for all real `x ,w h e r ef(x),g(x),a n dh(x)` are differentiable functions. At some point `x_0,F^(prime)(x_0)=21 F(x_0),f^(prime)(x_0)4f(x_0),g^(prime)(x_0)=-7g(x_0),` then the value of `g^(prime)(1)` is ________

Text Solution

Verified by Experts

The correct Answer is:
`(24)`
Given, `F(x) = f(x)* g(x)*h (x)`
On differentiating at `x = x_(0)`, we get
`F'(x_(0)) = f'(x_(0))* g(x_(0)) h(x_(0))+ f(x_(0))*g'(x_(0)) h(x_(0))+f(x_(0)) g (x_(0))h'(x_(0))` ...(i)
where, `F'(x_(0))=21 F(x_(0)),f'(x_(0)) = 4 f(x_(0)) g'(x_(0)) =- 7 g(x_(0)) and h'(x_(0)) and h'(x_(0)) = k h(x_(0)) `
On substituting in Eq. (i), we get
`21 F(x_(0))=4 f(x_(0)) g(x_(0)) h (x_(0)) -7 f(x_(0)) g(x_(0)) h(x_(0)) + k f(x) g(x_(0)) g(x_(0)) h(x_(0)) `
`rArr 21 = 4 -7 +k,["using F" (x_(0)) = f(x_(0)) g(x_(0)) h(x_(0))]`
` k = 24`
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