f(x) = 4x^3 + 3x^2 -6x +2
max = +infinity
min= -infinity
because it is cubic function and it is continuous and its domain and range is RxR.
f '(x) = 12 x^2 + 6 x - 6
f ''(x) = 24x + 6
To check concavity,
24x + 6 =0 => 6(4x +1) = 0 => 4x+1 =0 or x=-1/4
On putting value slightly lesser than -1/4 in f''(x) we get:
24x + 6
or 24(-1) +6
or -24+6
or -18 < 0
so from (-inf, -1/4) the function is concave downward
Now putting slightly greater value than -1/4 in f''(x) we get:
24(0) +6
or 6 > 0
so from (-1/4, inf) the function is concave upward
And becaues the f''(x) changes sign at -1/4 so -1/4 is an inflection point