### Home > CALC > Chapter 6 > Lesson 6.5.1 > Problem 6-171

Thoroughly investigate the graph of *f*(*x*) = *x*^{4/3} + 2*x*^{1/3} . Identify all important qualities, such as where the function is increasing, decreasing, concave up, and concave down. Also identify any point(s) of inflection and intercepts, and provide graphs of *f* ′(*x*) and *f* ″(*x*). Be sure to justify all statements both *graphically* and *analytically*. Homework Help ✎

Remember to investigate all extrema CANDIDATES. That means, find all values of *x* where *f*'(*x*) = 0 AND *f* '(*x*) DNE.

After all, a max or min could exist at a cusp... But remember, sometimes these candidates turn out to be inflection points instead.

When looking for points of inflection, remember to consider where concavity changes. This could happen where *f* ''(*x*) = 0

or where *f* ''(*x*) = DNE.