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For a function f integrable on [a,b], its average value on [a.b], also called its mean, is given by the formula av(f)- f(x) dx. Do f and the constant function av(f) b-a have the same integral over [a,b]? That is, does l av(f) dx = f(x) dx? Give reasons for your answer. Do f and av(f) have the same integral over [a.b]? Choose the correct answer below. O A. Yes, the function fmust equal av() for at least one value of x in [a,b] OB. Yes, let Jik) dx -A. Then av(Te integral of this constant over la.b) is A. O C. No, only when f is equal to the constant function av f) are the two integrals equal b-a No, sincavJ x) dx, and the integral of f over [a.b] is Jf(x) dx, the two integrals differ by a factor of -a
Show transcribed image text For a function f integrable on [a,b], its average value on [a.b], also called its mean, is given by the formula av(f)- f(x) dx. Do f and the constant function av(f) b-a have the same integral over [a,b]? That is, does l av(f) dx = f(x) dx? Give reasons for your answer. Do f and av(f) have the same integral over [a.b]? Choose the correct answer below. O A. Yes, the function fmust equal av() for at least one value of x in [a,b] OB. Yes, let Jik) dx -A. Then av(Te integral of this constant over la.b) is A. O C. No, only when f is equal to the constant function av f) are the two integrals equal b-a No, sincavJ x) dx, and the integral of f over [a.b] is Jf(x) dx, the two integrals differ by a factor of -a