The function $f (x) = x^{5} - 5 x^{4} + 5 x^{3} - 1$ has

one minimum and two maxima,

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two minima and one maximum,

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two minima and two maxima,

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one minimum and one maximum.

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Solution

The correct option is B one minimum and one maximum.
$f (x) = x^{5} - 5 x^{4} + 5 x^{3} - 1$
$f^{^{'}} (x) = 5 x^{4} - 20 x^{3} + 15 x^{2} = 5 x^{2} (x - 1) (x - 3) = 0$
$\Rightarrow x = 0, 1, 3$
$f^{^{''}} (x) = 10 x (2 x^{2} - 6 x + 3)$
now, $f^{^{''}} (0) = 0$
and $f^{^{''}} (1) = 10 (2 - 6 + 3) < 0$
and $f^{^{''}} (3) = 10 (18 - 12 + 3) > 0$
Therefore, $f (x)$ is minimum at $x = 3$ and maximum at $x = 1$
Ans: D

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The function f(x)=x5−5x4+5x3−1 has

The correct option is B one minimum and one maximum.f(x)=x5−5x4+5x3−1f′(x)=5x4−20x3+15x2=5x2(x−1)(x−3)=0⇒x=0,1,3f′′(x)=10x(2x2−6x+3)now, f′′(0)=0and f′′(1)=10(2−6+3)<0and f′′(3)=10(18−12+3)>0Therefore, f(x) is minimum at x=3 and maximum at x=1Ans: D

The function $f (x) = x^{5} - 5 x^{4} + 5 x^{3} - 1$ has