Question

How do you find the maximum, minimum and inflection points and concavity for the function `g(x) = 170 + 8x^3 + x^4`?

Answer 1

Point of minima `x=-6`, points of inflection `x=0, x=-4` & the function is concave in `x\in(-6, 0)`

Explanation:

The given function:

`g(x)=170+8x^3+x^4`

`g'(x)=24x^2+4x^3`

`g''(x)=48x+12x^2`

For mamima or minima we have

`g'(x)=0`

`24x^2+4x^3=0`

`4x^2(6+x)=0`

`x=0, -6`

Now, `g''(0)=48(0)+12(0)^2=0`

hence, `x=0` is a point of inflection

Now, `g''(-6)=48(-6)+12(-6)^2`

`=144>0`

hence, `x=-6` is a point of maxima

Now, for points of inflection we have

`g''(x)=0`

`48x+12x^2=0`

`12x(4+x)=0`

`x=0, -4`

The curve will be concave iff

`g'(x)<0`

`24x^2+4x^3<0`

`4x^2(6+x)<0`

`x\in(-6, 0)`

Thus, the graph will be concave for `x\in(-6, 0)`