Suppose f is integrable on [-b, b] and f is an odd function-that is, f(-t) = -f(t) for all tE(-b, b]. Prove that f, f dx = 0. If f is even-that is, f(-t) = f(t) for all t E [-b,bl– prove that f dx = 2 f dx.

Q: P 2. ( a) Find the average value of f on [9,25]. b) Find all c in [9, 25] such that f(c) = favg.…

A:

Q: 6. Let f be continuous on [a, b]. Show ffdF = f(x) [F(b) – F(a)] for some x € [a, b].

A: This is a consequence of the intermediate value theorem

Q: Find F as a function of x and evaluate it at x = 0, x = /3, and x = = Tt/2. F(x) 6 de

A: We will use basic formula of Integration for first one and from second part just Value of the…

Q: Let f be a real-valued function continuous at the point a = (a1, a2) in R². Keep a2 fixed and define…

A: Given that f is real valued function and continuous at a=a1,a2 in ℝ2 and gx=fx,a2. To show : g is…

Q: Let f be a real-valued function on [a,b] that is differentiable on (a,b). Suppose f '(x) < 0 for all…

A: This is a problem of the application of differential calculus.