Question

Suppose f, g are continuous and differentiable on [0,b], are non-negative on [0,b] and f is non constant with f(0) = 0, then the minimum value of ${\int }_0^{a}g\left(x\right)f^{\prime }\left(x\right)dx+{\int }_0^b{g}^{\prime }\left(x\right)f\left(x\right)dxonaϵ(0,b]$ is

• A. f(a).g(a)
• B. f(b).g(b)
• C. f(a).g(b)
• D. f(b).g(a)

Answer 1

The correct option is C f(a).g(b)

Integrating by parts the integral
${\int }_0^{a}g\left(x\right)f^{\prime }\left(x\right)dx+{\int }_0^b{g}^{\prime }\left(x\right)f\left(x\right)dx=g\left(x\right)f\left(x\right){|}_0^a-{\int }_0^a{g}^{\prime }\left(x\right)f\left(x\right)dx+{\int }_0^b{g}^{\prime }\left(x\right)f\left(x\right)dx=f\left(a\right).g\left(a\right)+{\int }_a^b{g}^{\prime }\left(x\right)f\left(x\right)dx\ge f\left(a\right).g\left(a\right)+{\int }_a^b{g}^{\prime }\left(x\right)f\left(x\right)dx=f\left(a\right).g\left(b\right)$