# Let f(x)=x^4-4x^3-144x^2+3x+2. What values are concave up,down and the inflection pointsVerify these please =)   I got this I found second dervative which is12(x^2-2x-24) 0=12(x+4)(x-6)   x=-4,...

Let f(x)=x^4-4x^3-144x^2+3x+2. What values are concave up,down and the inflection points

I got this

I found second dervative which is

12(x^2-2x-24)

0=12(x+4)(x-6)

x=-4, x=6

at x<-4 and x>6, it is concave up

at -4<x<6 it is concave down

Inflection points are -4 and 6

------------
Thank you

sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted on

You need to find first derivative of f(x), then, differentiating f'(x) yields the second derivative such that:

`f'(x) = 4x^3 - 12x^2 - 288x + 3`

`f"(x) = (4x^3 - 12x^2 - 288x + 3)'`

`f"(x) = 12x^2 - 24x - 288`

You need to factor out 12 such that:

`f"(x) = 12(x^2 - 2x - 24)`

Notice that the equation of second derivative is like the one that you have got, hence so far so good.

If you need to find the inflections of function, you should solve the equation f"(x) = 0 such that:

`x^2 - 2x - 24 = 0`

You should use quadratic formula such that:

`x_(1,2) = (2+-sqrt(4+96))/2 =gt x_(1,2) = (2+-sqrt100)/2`

`x_(1,2) = (2+-10)/2 =gt x_1 = 6; x_2 = -4`

Hence, the inflections of function occur at x=-4 and x=6.

You need to select a value between -4 and 6 to calculate the value of second derivative such that:

`x = 0 =gt f"(0) = -24`

Hence, the values of second derivative at values of x in (-4,6) are negative, hence the graph of function is concave down.

You need to select a value larger than 6, x=7 to calculate f"(7) such that:

`f"(7) = 49-14-25 = 10 gt 0`

Hence, the values of second derivative at values of x larger than 6 are positive, hence the graph of function is concave up.

You need to select a value smaller than -4, x=-5 to calculate f"(-5) such that:

`f"(-5) = 25 + 10 - 24 = 11 gt 0`

Hence, the values of second derivative at values of x smaller than -4 are positive, hence the graph of function is concave up.

Hence, the inflections of function occur at x=-4 and x=6 and the graph of function is concave up over `(-oo,-4)U(6;oo)`  and it is concave down over `(-4,6).`