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  4. The function f(x) = x5 - 5x4 + 5x3 - 1 has

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

Application of Derivatives

Solution:

$f(x) = 5x^4 - 20x^3 + 15x^2$
$f''(x) = 20x^3 - 60x^2 + 30\,x$
$f'(x) = 0$
$\Rightarrow 5x^2(x^2 - 4x + 3) = 0$
$\Rightarrow x = 0$, $1$, $3$
$f''(1) = -10 < 0$ and
$f''(3) = 540 - 540 + 90 = 90 > 0$
$f''(0) = 0$ and $f'''(0) \ne 0$
So, $x = 0$ is a point of inflexion.
$\therefore $ maximum at $x = 1$,
minimum at $x = 3$.