Question

Question 1. Consider the following claims: (1) So g(x) f(x)dx = SØg(x)dx · SOF(x)dx holds for all f and g continuous. (2) If f(x) is such that So f(x)dx = 0 then f(x) = 0 for all 0 < x < 1. (4) If f is continuous in [0, 1], then is bounded, independently of the value of n. The sum of all correct statements equals:

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Answer 1

I D false. and b. glas ad naredo cel, Fox) an Then, b I g(x). Jo) da = 3 x²da 19 17 ada g (al dan Flada 17 . Both are not neccessarily equal false. Let Then, Ild) a j glasda nd - Zie. f(x) to for ola?! ( ₂2-x) dx 6 - = - 0+0) But fla) to for oarel

4) True. if f is continuous in on Cours Then, it must have a finite area between It's graph and the noanis. Suppose we are estimating that area using in redan sub-intervals of equal width. Then, the width of each sub-interval is, ana -o - ñ and the value of function at the the in ch the right peoda endpoint of ith subinterval is ith subinsternal = cax fl ) fli of area -P À I LA .: Total area stils SW.5 Now, must . the area is finite, so the estimate also be finite and hence bounded. & 711) is bounded. DO

So the sum of all correct statements equals 4.

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