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∀x∈R ⇒h(x) is a constant function. ⇒h(10)=h(3) =(f(3))2−(ϕ(3))2 =(f(3))2−(f′(3))2 =52−42=9