Q47 / 120 · VITEEE 2016 · Mathematics

If $f(x)=x^{2}, g(x)=2 x, 0 \leq x \leq 2$ then the value of $\mathrm{I}(x)=\int_{0}^{2} \max (f(x), g(x))$ is

  • A
    $\frac{10}{3}$
  • B
    $\frac{1}{3}$
  • C
    $\frac{11}{3}$
  • Correct answer
    D
    32

Solution

\begin{array}{|c|c|c|} \hlinex & 0 & 2 \\ \hliner(x) & 0 & 16 \\ \hline \end{array} Let $r(x)=f(x) \cdot g(x)$ $=x^{2} \cdot 2 x=2 x^{3}$ $r^{\prime}(x)=6 x^{2}$ Put $6 x^{2}=0, \therefore x=0$ $\operatorname{Max} r(x)=2(2)^{3}=16$ or Max $(f(x), g(x))=16$ $I(x)=\int_{0}^{2} 16 d x$ $I(x)=[16 x]_{0}^{2}=32-0=32$

Hence, the correct answer is option: 'D'.

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