f(x) = 4x3 + 3x2 − 6x + 2

Question

find max and min as well as inflection point. concaves up? concaves down?

Answer 1

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