• Problem 6. Let I CR be an interval. Let 0:1 → R be differentiable. Let CER be such that for all t El, be such that “O

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Problem 6 Let I Cr Be An Interval Let 0 1 R Be Differentiable Let Cer Be Such That For All T El Be Such That O 1 (87.57 KiB) Viewed 26 times

• Problem 6. Let I CR be an interval. Let 0:1 → R be differentiable. Let CER be such that for all t El, be such that “O <co(t). (1) Define f : 1 + R as f(w) := e-cwo(t) for (WEI). Show that f'(w) < 0 for all wel. Conclude that f is decreasing on I. (2) Define g: 1 + R as g(w) := ecu o(t) for (w EI). Show that g'(w) > 0 for all w El. Conclude that g is increasing on I. (3) Show for all t, s e I satisfying s < t, that o(t) <eclt-slo(s). (Hint: Use that f is decreasing (4) Show for all t, s e I satisfying t <s, that o(t) <eclt-10(s). (Hint: Use that g is increasing|- (5) Show for all t, s e I satisfying t = s, that o(t) <ect-s10(s). (Hint: This is not complicated, so don't overthink it).