answer:Since f(ab)=f(a)+f(b), We have f(a^{2})=f(a)+f(a)=2f(a), And f(a^{3})=f(a^{2}⋅a)=f(a^{2})+f(a)=2f(a)+f(a)=3f(a), Given that f(2)=q and f(3)=p, We can find f(72) as follows: f(72)=f(9×8)=f(3^{2}⋅2^{3})=f(3^{2})+f(2^{3})=2f(3)+3f(2)=boxed{2p+3q}, Hence, the answer is C. According to the problem, we can find that f(72)=3f(2)+2f(3), which leads to the answer. This problem tests the understanding of abstract functions and their applications. Obtaining f(a^{2})=2f(a) and f(a^{3})=3f(a) from f(ab)=f(a)+f(b) is crucial, and this problem assesses the ability to transform and perform calculations. It is of moderate difficulty.

answer:The function is given by g(x) = 3x - 7. To find the inverse, we solve for x: [ y = 3x - 7 ] [ y + 7 = 3x ] [ x = frac{y + 7}{3} ] Thus, g^{-1}(x) = frac{x + 7}{3}. We want to find the value of x where g(x) = g^{-1}(x): [ 3x - 7 = frac{x + 7}{3} ] Multiplying through by 3 to clear the fraction: [ 9x - 21 = x + 7 ] [ 8x = 28 ] [ x = frac{28}{8} = boxed{3.5} ]

answer:1. Given the polynomial ( f(x) = ax^2 + bx + c ) with the constraints ( f(0) = 0 ) and ( f(2) = 2 ), we need to find the minimum value of ( S = int_0^2 |f'(x)| , dx ). 2. From the constraint ( f(0) = 0 ), we have: [ f(0) = a(0)^2 + b(0) + c = 0 implies c = 0 ] Thus, the polynomial simplifies to: [ f(x) = ax^2 + bx ] 3. From the constraint ( f(2) = 2 ), we have: [ f(2) = a(2)^2 + b(2) = 2 implies 4a + 2b = 2 implies 2a + b = 1 ] 4. We need to find the minimum value of ( S = int_0^2 |f'(x)| , dx ). First, we compute ( f'(x) ): [ f'(x) = frac{d}{dx}(ax^2 + bx) = 2ax + b ] 5. Using the inequality ( int_0^2 |f'(x)| , dx geq left| int_0^2 f'(x) , dx right| ), we compute: [ int_0^2 f'(x) , dx = int_0^2 (2ax + b) , dx ] 6. Evaluating the integral: [ int_0^2 (2ax + b) , dx = left[ ax^2 + bx right]_0^2 = (4a + 2b) - (0 + 0) = 4a + 2b ] 7. From the constraint ( 2a + b = 1 ), we substitute ( b = 1 - 2a ) into ( 4a + 2b ): [ 4a + 2(1 - 2a) = 4a + 2 - 4a = 2 ] 8. Therefore, we have: [ left| int_0^2 f'(x) , dx right| = |4a + 2b| = |2| = 2 ] 9. Since ( int_0^2 |f'(x)| , dx geq left| int_0^2 f'(x) , dx right| ), the minimum value of ( int_0^2 |f'(x)| , dx ) is: [ int_0^2 |f'(x)| , dx geq 2 ] 10. The minimum value occurs when ( f'(x) geq 0 ) for all ( x in [0, 2] ). Given ( f'(x) = 2ax + b ) and substituting ( b = 1 - 2a ), we get: [ f'(x) = 2ax + (1 - 2a) = 2ax + 1 - 2a ] For ( f'(x) geq 0 ) for all ( x in [0, 2] ), we need: [ 2ax + 1 - 2a geq 0 implies 2ax geq 2a - 1 implies x geq frac{2a - 1}{2a} ] For ( x in [0, 2] ), this inequality must hold for ( x = 0 ): [ 0 geq frac{2a - 1}{2a} implies 2a - 1 leq 0 implies a leq frac{1}{2} ] And for ( x = 2 ): [ 2a(2) + 1 - 2a geq 0 implies 4a + 1 - 2a geq 0 implies 2a + 1 geq 0 implies a geq -frac{1}{2} ] Therefore, ( -frac{1}{2} leq a leq frac{1}{2} ). 11. The minimum value of ( int_0^2 |f'(x)| , dx ) is achieved when ( a = frac{1}{2} ) and ( b = 0 ), giving: [ f(x) = frac{1}{2}x^2 ] [ f'(x) = x ] [ int_0^2 |f'(x)| , dx = int_0^2 x , dx = left[ frac{x^2}{2} right]_0^2 = 2 ] Conclusion: [ boxed{2} ]

answer:1. **Identify the highest power of x**: here it is x^5. 2. **Calculate the coefficient of x^5:** - From 5(x^5 - x^4 + x^3), the coefficient of x^5 is 5. - From -8(x^5 + 2x^3 + 1), the coefficient of x^5 is -8. - From 6(3x^5 + x^4 + x^2), the coefficient of x^5 is 6 times 3 = 18. - Add these coefficients: 5 + (-8) + 18 = 15. 3. **Conclusion**: The leading coefficient of the polynomial is boxed{15}.