Q: Let a + b = 4, where a < 2, and let g(x) be a differentiable function. If dg/dx > 0 for all x , Prove that $\displaystyle \int_{0}^{a} g(x) dx + \int_{0}^{b} g(x) dx $ increases as (b – a) increases. Sol. Let b – a = t given a + b = 4 so a = 2 -t/2 , b = 2 + t/2 Let $\displaystyle f(t) = \int_{0}^{a} g(x) dx + \int_{0}^{b} g(x) dx $ $\displaystyle f(t) = \int_{0}^{2-\frac{t}{2}} g(x) dx + \int_{0}^{2+\frac{t}{2}} g(x) dx $ $\displaystyle f'(t) = g(2-\frac{t}{2})(-\frac{1}{2}) + g(2+\frac{t}{2})(\frac{1}{2}) $ $\displaystyle f'(t) = \frac{1}{2}[g(b)- g(a)]$ Since $\frac{dg}{dx} > 0$ for all x, so g(x) is increasing since b > a g(b) > g(a) Hence, f'(t) > 0 f(t) increasing as t increases i.e. f(t) increases as (b – a) increases Also Read :The function $(x^2 - 1)|x^2 - 3x + 2| + cos|x|$ is not differentiable atf : [1 , ∞) → R : f(x) is a monotonic and differentiable function and f(1) = 1, then number of…The function $ f (x) = frac{x^2 - 3x + 2}{x^2 + 2x - 3} $ isLet f : [-π/3 , 2π/3] → [0 , 4] be a function defined as f(x)= sin x – cos x + 2 . Then f -1(x) is…The range of the function $ y = sqrt{-2 cos^2 x + 3 cosx - 1}$ isDomain of the function $f(x) = frac{x}{sqrt{sin(lnx)-cos(lnx)}}$ is$ f(x) = [tan^2 x]$ where [.] denotes the greatest integer function. ThenThe function $f(x) = frac{log(1 + ax)-log(1 - bx)}{x} $ is not defined at x= 0. The value which…f(x) = tan^-1 (sinx + cosx) is an increasing function inIf f' (x) = 1-2sin^2 x/f(x) , ( f(x) ≥ 0, ∀ x ∈ R and f(0) = 1) then f (x) is a periodic function… Post navigation A conical vessel is to be prepared out of a circular sheet of gold of unit radius. How much sectorial area is to be removed …. $ Let \; f(x) = \left\{\begin{array}{ll} -x^3 + \frac{b^3 – b^2 + b – 1}{b^2 + 3b +2} \; , 0 \leq x < 1 \\ 2x - 3 \; , 1 \leq x \leq 3 \end{array} \right. $