Question

How do you determine whether the function `f(x)=x^8(ln(x))` is concave up or concave down and its intervals?

Answer 1

See the explanation.

Explanation:

The intervals of concavity are determined by 2nd derivative. If 2nd derivative changes sign in the points where it is equal to zero, those points are inflection points.

`D_f=R^+`

`f'(x)=8x^7lnx+x^8 1/x=8x^7lnx+x^7=x^7(8lnx+1)`

`f''(x)=7x^6(8lnx+1)+x^7 8/x=7x^6(8lnx+1)+8x^6`

`f''(x)=x^6(56lnx+15)`

`f''(x)=0 <=> x^6(56lnx+15)=0 <=>`

`<=> (x^6=0 vv 56lnx+15=0)`

`x^6=0 <=> x=0 !in D_f`

`56lnx+15=0 <=> 56lnx=-15 <=> lnx=-15/56 <=>`

`<=> x=e^(-15/56) <=> x=1/root(56)(e^15) ~~ 0.765>0 in D_f`

`AAx in (0,1/root(56)(e^15)): f''(x)<0` function is concave down

`AAx in (1/root(56)(e^15), oo): f''(x)>0` function is concave up

Note: the sign of `f''(x)` depends only on `56lnx+15` since `x^6` is always greater then zero for `x>0`.