EE 518 : Homework #6 Due on Monday 2:00pm, October 10, 2016Problem 1Suppose f, g : [a, b] → R is differentiable on its domain, and g06= 0. Show that ∃c ∈ (a, b), such thatf(a) − f(c)g(c) − g(b)=f0(c)g0(c)Hint: Construct proper function to apply mean value theoremProblem 2If f(x) > g(x) on [a, b], and f, g are Riemann integrable on [a, b] show thatZbaf(x)dx >Zbag(x)dxHint: Use definition of Riemann integral.Problem 3Let f : (0, ∞) → < denote the Gamma Function, i.e.f(α) =Z∞0xα−1e−xdx(i) Show that f(α) is well defined for any α > 0, i.e., show that ∀α > 0, bothZ10xα−1e−xdx = limt→0Z1txα−1e−xdxandZ∞1xα−1e−xdx = limt→∞Zt1xα−1e−xdxexists.(ii) Show thatf(α) = (α − 1)f (α − 1), ∀α > 1and consequently,f(n) = (n − 1)!, ∀n ∈ Z+Problem 4Calculate the following integrals(i)Ze1(x ln x + xex)(ii)Zπ/20sinnxdx, n = 0, 1, 2, . . .(hint: consider both n is even and odd number)Page 1 of 2EE 518 : Homework #6 Due on Monday 2:00pm, October 10, 2016Problem 5Suppose a function f is continuous on [a, c], and differentiable on (a, c).Rbaf(x)dx =Rcbf(x)dx = 0, forsome b ∈ (a, c). Show that there exists a ξ ∈ (a, c), such that: f0(ξ) = 0Problem 6For the following parts, submit your code and your results:1. CalculateRsin(x) + x cos(x)dx2. CalculateR10α√x − αdx3. CalculateR100xx2+5dx using some numerical methodPage 2 of