ANSWER
Relative minimum occurs at x=0
Relative maximum occurs at x=-2
EXPLANATION
The given function is;
[tex]f(x) = 3 {x}^{3} + 9 {x}^{2} - 1[/tex]
We take the first derivative to get:
[tex]f'(x) = 9 {x}^{2} + 18x[/tex]
At turning point f'(x)=0
[tex] 9 {x}^{2} + 18x = 0[/tex]
[tex] 9x(x + 2) = 0[/tex]
[tex]x = 0 \: or \: x = - 2[/tex]
We take the second derivative to get
[tex]f''(x) = 18x + 18
[/tex]
[tex]f''(0) = 18(0) + 18 = 18 \: > \: 0[/tex]
Relative minimum occurs at x=0
[tex]f''( - 2) = 18( - 2)+ 18 = - 18 \: < \: 0[/tex]
Relative maximum occurs at x=-2
