Question

What are the minimum and maximum values of the function

$x^{5} - 5x^{4} + 5x^{3} - 10$

• A. $- 37, - 9$
• B. 10, 0
• C. It has 2 minimum and 1 maximum values
• D. It has 2 maximum and 1 minimum values

Answer 1

Answer: (A)

$- 37, - 9$

Answer 2

$y = x^{5} - 5x^{4} + 5x^{3} - 10$

$\therefore\frac{dy}{dx} = 5x^{4} - 20x^{3} + 15x^{2} = 5x^{2}(x^{2} - 4x + 3)$

=$5x^{2}(x - 3)(x - 1)$

$\frac{dy}{dx} = 0$, gives $x = 0,1,3$ ......(i)

Now, $\frac{d^{2}y}{dx^{2}} = 20x^{3} - 60x^{2} + 30x = 10x(2x^{2} - 6x + 3)$ and

$\frac{d^{3}y}{dx^{3}} = 10(6x^{2} - 12x + 3)$

For $x = 0$:$\frac{dy}{dx} = 0,\frac{d^{2}y}{dx^{2}} = 0,\frac{d^{3}y}{dx^{3}} \neq 0$,

$\therefore$ Neither minimum nor maximum

For $x = 1$, $\frac{d^{2}y}{dx^{2}} = - 10$=negative,

$\therefore$ Maximum value $y_{max.}$

For $x = 3$, $\frac{d^{2}y}{dx^{2}} = 90$=positive,

$\therefore$ Minimum value $y_{min.}$.