What was found
FL350, THRUST SET TO CLIMB, NR 1 ITT YELLOW RANGE AT APPROX 903-904 CELSIUS. POWER RETARDED & ITT REDUCED TO GREEN. ITT REMAINED 80-90 DEGREES HOTTER THAN NR 2 ENG ITT, A HIGHER FUEL FLOW OBSERVED, & VIBRATIONS ON HP SPOOL OF NR 1 NOTABLY HIGHER (STILL WITHIN LIMITS) THAN VIBRATIONS ON NR 2. OIL INDS ON TAKEOFF BOTH NORMAL, & NO MESSAGES PRESENTED. PERFORMED PRECAUTIONARY ENG SHUTDOWN & LANDED AT NEAREST AIRPORT. ATC NOTIFIED. PRECAUTIONARY ENGINE SHUTDOWN PROCEDURE FOLLOWED. ATTEMPTED APU START, FAILED TO ACCELERATE. APPEARED IT WAS NOT LIGHTING PROPERLY, NO RISING EGT NOTED. ACCOMP, & AT LOWER ALTITUDES & WITHIN LIMITS OF APU STARTER, ATTEMPTED TO START 2 TIMES, SAME RESULT. LANDED W/O INCIDENT.