The given function f(x) = x⁵ – 5x⁴ + 5x³ – 1; has/have

(a) local maxima at x = 1
(b) local maximum value is 0
(c) local minimum at x = 3
(d) local minimum value is –28
(e) The point of inflexion is x = 1

Choose the correct answer from the options given below

(a), (b), (c), (d) only

$f(x)= x^5-5 x^4+5 x^3-1$

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

$\Rightarrow 5 x^2\left(x^2-4 x+3\right)$

$\Rightarrow 5 x^2(x-1)(x-3)$

so  $f'(x) = 0$

$\Rightarrow x = 0, 1 , 3$

$f''(x)=20 x^3-60 x^2+30 x=10 x\left(2 x^2-6 x+3\right)$

So $f''(0)=0$ (point of intersection)

$f''(1)=-10<0$   (point of maxima)

$f''(3)=10 \times 3(3)>0$   (point of minima)

$f(3)=3^5-5 \times 3^4+5 \times 3^3-1=-28$   (minimum value)

Option: 3