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Question

Let f and g be two differentiable functions such that f(x)=ϕ(x) and ϕ(x)=f(x) for all real x. If f(3)=5 and f(3)=4, then the value of (f(10))2(ϕ(10))2 is

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Solution

Let h(x)=(f(x))2(ϕ(x))2
h(x)=2f(x)f(x)2ϕ(x)ϕ(x)
h(x)=2f(x)f(x)2f(x)f(x)
(f(x)=ϕ(x), ϕ(x)=f(x))
h(x)=0 xR
h(x) is a constant function.
h(10)=h(3)
=(f(3))2(ϕ(3))2
=(f(3))2(f(3))2
=5242=9

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