- #1
geoffrey159
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Homework Statement
Let ##f,g## be two real valued functions, defined on the segment ##[a,b]## and continuous on ##[a,b]##, such that ## 0 < g < f ##. Show there exist ##\lambda > 0 ## such that ## (1+\lambda) g \le f ##
Homework Equations
The Attempt at a Solution
Set ##h = f/g##. Since ##g\neq 0##, ##h## is continuous on ##[a,b]##.
Therefore, ##h## is bounded on ##[a,b]## and reaches its bounds. Call ##m = h(x_0)## its lower bound. By construction, ##1 < m \le h##, so ## \lambda = m-1 > 0 ## and ## (1+\lambda) g \le f ##. Is it correct ?