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If $f (x) = x^{5} - 5 x^{4} + 5 x^{3} - 10$ has local max. and min. at $x = a$ and $x = b$ respectively, then $(a, b)$ is

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$(0, 1)$

$(1, 3)$

$(1, 0)$

$(3, 0)$

answer is B.

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Detailed Solution

$f' (x) = 5 x^{4} - 20 x^{3} + 15 x^{2} = 0$

$\Rightarrow 5 x^{2} (x^{2} - 4 x + 3) = 0 \Rightarrow x = 0$

$\Rightarrow f'' (x) = 20 x^{3} - 60 x^{2} + 30 x$

$f'' (1) = 20 - 60 + 30 < 0, f' (3) > 0$

$\Rightarrow (a, b) = (1, 3)$

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