Let f be a real valued continuous function defined on the positive real axis such that g(x)=∫ tf(t)dt t ∈ to (0, x).

Question

Let f be a real valued continuous function defined on the positive real axis such that \(\mathrm{g}(\mathrm{x})=\int\limits_{0}^{\mathrm{x}} \mathrm{t} f(\mathrm{t}) \mathrm{dt}\) If \(g\left(x^{3}\right)=x^{6}+x^{7},\) then value of \(\sum\limits_{r=1}^{15} f\left(r^{3}\right)\) is :

(1) 320

(2) 340

(3) 270

(4) 310

Answer 1

Correct option is (4) 310  

\(g(x)=x^2+x^{\frac{7}{3}}\)

\(g^{\prime}(x)=2 x+\frac{7}{3} x^{4 / 3} \)

\( f(x)=\frac{g^{\prime}(x)}{x} \)

\( f(x)=2+\frac{7}{3} x^{1 / 3} \)

\( f\left(r^3\right)=2+\frac{7}{3} r \)

\(\sum\limits_{r=1}^{15}\left(2+\frac{7}{3} r\right)=2(15)+\frac{7}{3}\left(\frac{15(16)}{2}\right)\)     

= 310