# f(x)= xsqrt((x^2)+16) is defined on the interval [-4,6] A. f(x) is concave down on the region ? B. f(x) is concave up on the region  ? C. The inflection point for this function is at x=?

`f(x)=x*sqrt(x^2+16)=x*(x^2+16)^(1/2)=>`

`f'(x)=1*(x^2+16)^(1/2)+x*1/2*(x^2+16)^(-1/2)*2x=>`

`f'(x)=sqrt(x^2+16)+x^2/sqrt(x^2+16)`

`f''(x)=1/2*(x^2+16)^(-1/2)*2x+[2x*sqrt(x^2+16)-x^2*1/2*(x^2+16)^(-1/2)*2x]/[x^2+16]=>`

`f''(x)=x/sqrt(x^2+16)+[2x*(x^2+16)-x^3]/[sqrt(x^2+16)*(x^2+16)]`

`f''(x)=[x*(x^2+16)+2x*(x^2+16)-x^3]/[sqrt(x^2+16)(x^2+16)]=>`

`f''(x)=[16x+2x^3+32x]/[sqrt(x^2+16)(x^2+16)]=>`

`f''(x)=[48x+2x^3]/[sqrt(x^2+16)(x^2+16)]`

Denominator is always positive =>

`f''(x)<0=>48x+2x^3>0=>`

`2x(24+x^2)>0=>2x>0=>x>0`

f(x) concave down when f''(x)<0=>concave down on `[-4,0)`

f(x) concave up when f''(x)>0=> concave up on `(0,6]`

Thus point of inflection at x=0

`f(x)=x*sqrt(x^2+16)=x*(x^2+16)^(1/2)=>`

`f'(x)=1*(x^2+16)^(1/2)+x*1/2*(x^2+16)^(-1/2)*2x=>`

`f'(x)=sqrt(x^2+16)+x^2/sqrt(x^2+16)`

`f''(x)=1/2*(x^2+16)^(-1/2)*2x+[2x*sqrt(x^2+16)-x^2*1/2*(x^2+16)^(-1/2)*2x]/[x^2+16]=>`

`f''(x)=x/sqrt(x^2+16)+[2x*(x^2+16)-x^3]/[sqrt(x^2+16)*(x^2+16)]`

`f''(x)=[x*(x^2+16)+2x*(x^2+16)-x^3]/[sqrt(x^2+16)(x^2+16)]=>`

`f''(x)=[16x+2x^3+32x]/[sqrt(x^2+16)(x^2+16)]=>`

`f''(x)=[48x+2x^3]/[sqrt(x^2+16)(x^2+16)]`

Denominator is always positive =>

`f''(x)<0=>48x+2x^3>0=>`

`2x(24+x^2)>0=>2x>0=>x>0`

f(x) concave down when f''(x)<0=>concave down on `[-4,0)`

f(x) concave up when f''(x)>0=> concave up on `(0,6]`

Thus point of inflection at x=0

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