# Convexity, concave

• Jan 2nd 2009, 09:14 AM
abstar
Convexity, concave
1-Over what domain the function \$\displaystyle f(x)=x^2(x^2-1)\$ strictly convex?
2-Give a linear and quadratic approximations for this function at the point \$\displaystyle x=1\$
3-Are these approximations convex, concave or neither?

Thank you very much.
• Jan 2nd 2009, 10:04 AM
Last_Singularity
Quote:

Originally Posted by abstar
1-Over what domain the function \$\displaystyle f(x)=x^2(x^2-1)\$ strictly convex?
2-Give a linear and quadratic approximations for this function at the point \$\displaystyle x=1\$
3-Are these approximations convex, concave or neither?

Thank you very much.

Convexity is determined by the sign of the second order derivative.

Linear and quadratic approximations can be obtained by finding the Taylor Series expansion of the function

Please try out the problem first and let us know where you get stuck.
• Jan 2nd 2009, 10:58 AM
abstar
What about the Hessian matris in looking for the convexity.
As far as i know, If the Hessian matrice is positive definite: the function is strictly convex, If it is positive semidefinite it is convex. And if the Hessian is negative than concave. Am i right?

Quote:

Originally Posted by Last_Singularity
Convexity is determined by the sign of the second order derivative.

Linear and quadratic approximations can be obtained by finding the Taylor Series expansion of the function

Please try out the problem first and let us know where you get stuck.