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
Answer & explanation
Correct answer: option 3
$f(x)= x^5-5 x^4+5 x^3-1$
$f'(x)= 5 x^4-20 x^3+15 x^2$
$\Rightarrow 5 x^2\left(x^2-4 x+3\right)$
$\Rightarrow 5 x^2(x-1)(x-3)$
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