Paragraph: Let f(x)=P e^x+Q e^ 2 x +R e^ 3 x , where P, Q, R are real numbers. Further f(0)=6, f^ ( 3)=282 and ₀^ 2 f(x) d x=11 Question: What is the value of Q ?

Step-by-step solution

f(x)=P e^x+Q e^ 2 x +R e^ J x f(0)=P+Q+R=6 ....(1) f^ (x)=P e^x+2 Q e^ 2 x +3 R e^ 3 x f^ ( 3)=P e^ 3 +2 Q e^ 2 3 +3 R e^ 3 3 3 P+18 Q+81 R=282 P+6 Q+27 R=94 ....(2) Since f(x) d x=P e^x+ Q 2 e^ 2 x + R 3 e^ 3 x ₀^ f(x) d x=11 (P e^ In 2 + Q 2 e^2 In 2+ R 3 e^3 In 2-P- Q 2 - R 3 )=11 2 P+2 Q+ 8 R 3 -P- Q 2 - R 3 =11 P+ 3 Q 2 + 7 R 3 =11 ....(3) (2) - (1) and (3) - (1), 5 Q+26 R=88 ....(4) and, Q 2 + 4 R 3 =53 Q+8 R=30 ....(5) From (5), R= 30-3 Q 8 From (4), 5 Q+26 ( 30-3 Q 8 )=88 20 Q+390-39 Q=352 19 Q=38 Q=2