Find the inflection points and discuss the concavity for the function `f(x)=xsqrt(x+3) `
Inflection points occur where the second derivative is zero (and changes sign.)
Rewrite as ` f(x)=x(x+3)^(1/2) `
Using the product rule we get:
Setting f''(x)=0 we get:
x=-3 or x=-4. Since x=-4 is not in the domain the only zero is x=-3. This is the endpoint of the domain, so there are no inflection points. The second derivative is positive on the functions domain, so the function is concave up everywhere.