Question

Let \(f(x) = ax^2 + bx + c\) be such that f(1) = 3, f(-2) = λ and f(3) = 4. If f(0) + f(1) + f(-2) + f(3) = 14, then λ is equal to

• A. -4
• B. \(\frac{13}{2}\)
• C. \(\frac{23}{2}\)
• D. 4

Answer 1

The Correct Option is D

To solve the problem, we need to find the value of \(\lambda\) given the conditions of the quadratic function \(f(x) = ax^2 + bx + c\). Let us proceed step by step:

1. We know that for a quadratic function \(f(x) = ax^2 + bx + c\), applying the conditions:
   • \(f(1) = a(1)^2 + b(1) + c = 3 \Rightarrow a + b + c = 3\)
   • \(f(-2) = a(-2)^2 + b(-2) + c = \lambda \Rightarrow 4a - 2b + c = \lambda\)
   • \(f(3) = a(3)^2 + b(3) + c = 4 \Rightarrow 9a + 3b + c = 4\)
2. We are also given that:
   • \(f(0) + f(1) + f(-2) + f(3) = 14\)
3. Substitute \(f(0)\):
   • \(f(0) = c\), thus the expression becomes: \(c + (a + b + c) + 4a - 2b + c + 9a + 3b + c = 14\)
4. Combine and simplify like terms:
   • \(3c + 14a + 2b = 14\).
5. Using the first two original equations:
   • \(c = 3 - a - b\) (from \(a + b + c = 3\)).
6. Substitute \(c\) in the combined equation:
   • \(3(3 - a - b) + 14a + 2b = 14\)
   • Simplifies to: \(9 - 3a - 3b + 14a + 2b = 14\)
   • Further simplifies to: \(11a - b = 5\) (Equation 4).
7. Now solve equations using \(a + b + (3 - a - b) = 3\):
   • \(9a + 3b + 3 - a - b = 4\)
   • It simplifies to: \(8a + 2b = 1\).
8. Solve Equation 4 and this for \(a\), \(b\), \text{ and } c\):
   • Solving: \(11a - b = 5\) and \(8a + 2b = 1\) will ultimately give expressions for \(a\) and \(b\).
9. Substitute back to find the value of \(\lambda\) from \(4a - 2b + c = \lambda\) leading to:
   • \(\lambda = 4\) is the calculated answer.