Let f:R→R be a function defined by f(x)=(2+3a)x2+(a−1a+2)x+b,a=1. If f(x+y)=f(x)+f(y)+1−72xy, then the value of 28∑i=15∣f(i)∣ is
Get the complete, step-by-step math solution for: "Let f: {R} {R} be a function defined by f(x) = (2 + 3a) x² + (a + 2)/(a - 1) x + b, a ≠ 1. If f(x + y) = f(x) + f(y) + 1 - (2)/(7) xy, then the value ...". Powered by SolveForX AI math tutor.
Step-by-Step Solution
Step 1: Analyze the given functional equation
The problem provides a functional equation relating f(x+y) to f(x) and f(y). This equation is key to determining the coefficients of the quadratic function f(x). We will substitute x=0 and y=0 into this equation to find the value of b. Then, we will substitute the general form of f(x) into the functional equation to find the values of a and b.
Step 2: Determine the constant term 'b'
Substitute x=0 and y=0 into the functional equation f(x+y)=f(x)+f(y)+1−72xy. This simplifies to f(0)=f(0)+f(0)+1, which implies f(0)=−1. From the definition of f(x), we know that f(0)=b. Therefore, we find that b=−1.
Step 3: Substitute f(x) into the functional equation
Now, substitute the expression for f(x) into the functional equation. Expand both sides and simplify. This will allow us to compare the coefficients of xy and other terms to find the value of a.
Step 4: Solve for 'a'
Expand the left side of the equation. After simplification, the terms involving x2, y2, x, and y cancel out on both sides. We are left with a comparison of the xy terms. Equating the coefficients of xy gives 2(2+3a)=−72. Solving this equation for a yields a=−75.
Step 5: Determine the final form of f(x)
Now that we have the values of a=−75 and b=−1, we can substitute them back into the original definition of f(x). Calculate the coefficients (2+3a) and (a−1a+2) to get the complete expression for f(x).
Step 6: Calculate ∑i=15∣f(i)∣
Now we need to calculate f(i) for i=1,2,3,4,5. Since all the coefficients of f(x) are negative and the constant term is negative, f(x) will be negative for positive values of x. Therefore, ∣f(i)∣=−f(i). Calculate each f(i) and sum their absolute values.
Step 7: Calculate the final value
Finally, multiply the sum of the absolute values by 28 to get the required answer.