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  • If  f is twice differentiable function such that <img alt="f^{\prime \prime}(x)=f(x)" src="https://en

If  f is twice differentiable function such that f^{\prime \prime}(x)=f(x), and f^{\prime}(x)=g(x), h(x)=[f(x)]^2+[g(x)]^2 and h(5)=11, then h(10)=

Option: 1

22


Option: 2

11


Option: 3

15


Option: 4

none


Answers (1)

best_answer

Differentiating the given relation h(x)=[f(x)]^2+[g(x)]^2

with respect to x, we get

h^{\prime}(x)=2 f(x) f^{\prime}(x)+2 g(x) g^{\prime}(x)

But we are given f^{\prime \prime}(x)=-f(x) and f^{\prime}(x)=g(x) so that f^{\prime \prime}(x)=g^{\prime}(x). Then (i) may be rewritten as

h^{\prime}(x)=-2 f^{\prime \prime}(x) f^{\prime}(x)+2 f^{\prime}(x) f^{\prime \prime}(x)=0

Thus h^{\prime}(x)=0,

Whence by integrating, we get

h(x)=constant=c , say 

Hence h(x)=c, for all x.

In particular, h(5)=c. But we are given h(5)=11.

It follows that c=11 and we have h(x)=11 for all x.

Therefore h(10)=11.

Posted by

Sanket Gandhi

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