Maths · Functions
If =a x^{2}+b x+c \) satisfies the identity -f(x)=8 x+3 \) for all Then = \)
If \( f(x)=a x^{2}+b x+c \) satisfies the identity \( f(x+1)-f(x)=8 x+3 \) for all \( x \in R . \) Then \( (a, b)= \)
- A. (2,1) (i)
- B. (4,-1)
- C. (-1,4)
- D. (-1,1)
Step-by-step solution
Compute f(x+1) = a(x+1)^2+b(x+1)+c = ax^2+2ax+a+bx+b+c. Then f(x+1)-f(x) = 2ax + (a+b). Set equal to 8x+3: 2a=8 => a=4; a+b=3 => b=-1. Thus (a,b)=(4,-1).
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