Continuous Functions with Constraints: f(x) > -1 and f(x) = x + ∫₁ˣf(t) dt

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SUMMARY

The discussion focuses on finding all continuous functions f(x) that satisfy the conditions f(x) > -1 and f(x) = x + ∫₁ˣ f(t) dt for all x. It is established that f must be twice differentiable, leading to the differentiation of the equation twice, resulting in f''(x) = f'(x). The solution to this differential equation reveals that f(x) can be expressed in terms of exponential functions, specifically f(x) = Ce^x + x - 1, where C is a constant determined by the constraint f(x) > -1.

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  • Knowledge of integral calculus, specifically definite integrals
  • Familiarity with differential equations and their solutions
  • Concept of differentiability and its implications in function behavior
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erogol
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Find all continuous functions f(x) satisfying
f(x) > −1 and [tex]f(x) = x + \int_{1}^{x} f(t) dt[/tex]


for all x
 
Physics news on Phys.org
First show f must be twice differentiable.
Next differentiate your equation twice to yield
f''=f'
find f
 

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