All are true about cephalosporins, except:
SECOND GENERATION CEPHALOSPORINS 
These were developed subsequent to the first generation compounds and are more active against gram-negative organisms with wider coverage, including some strains resistant to first generation compounds. Few members, i.e. cefoxitin are active against anaerobes as well, but none inhibits P. aeruginosa.
They are weaker than the first  generation compounds against gram positive bacteria.
Their utility has declined in favour of the 3rd generation agents.
Cefoperazone: Like ceftazidime, it differs from other third generation compounds in having stronger activity on Pseudomonas and weaker activity on other organisms. Cefoperazone is good for S. typhi and B.fragilis also, but more susceptible to beta-lactamases.
Ceftazidime: The most prominent feature of this third generation cephalosporin is its high activity against Pseudomonas aeruginosa, and the specific indications are- febrile neutropenic patients with haematological malignancies, burn, etc. lts activity against Enterobacteriaceae is similar to that of cefotaxime, but it is less active on Staph. aureus, other gram positive cocci and anaerobes like Bact. fragilis.
Reference: Essentials of Medical Pharmacology Eighth Edition KD TRIPATHI page no 777,778
Ceftazidime is a 3rd generation cephalosporin.
Cefoperazone has got antipseudomonal effect.
Cefoxitin has got no activity against anaerobes.
Cephalosporins act by inhibiting cell wall synthesis.