Indefinite Integration 3 Question 22
22. Investigate for maxima and minima the function,
$f(x)=\int _1^{x}\left[2(t-1)(t-2)^{3}+3(t-1)^{2}(t-2)^{2}\right] d t$.
$(1988,5$ M)
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Solution:
- Given, $f(x)=\int _1^{x}\left[2(t-1)(t-2)^{3}+3(t-1)^{2}(t-2)^{2}\right] d t$
$\therefore \quad f^{\prime}(x)=\left[2(x-1)(x-2)^{3}+3(x-1)^{2}(x-2)^{2}\right] \cdot 1-0$
$$ =(x-1)(x-2)^{2}[2(x-2)+3(x-1)] $$
$$ =(x-1)(x-2)^{2}(5 x-7) $$
$\therefore f(x)$ attains maximum at $x=1$ and $f(x)$ attains minimum at $x=\frac{7}{5}$.
