Question

Determine the coefficients a and b such that f(x)=x2+ax+b satisfies f(1)=5 and f ′(1)=1. What is the value of b?

A) b=2
B) b=3
C) b=4
D) b=5

Answer 1

To determine the coefficients 'a' and 'b' for the quadratic function f(x) = x^2 + ax + b using the given conditions f(1) = 5 and f'(1) = 1, we find that 'a' must be -1 and 'b' must be 5.

Step-by-step explanation:

A student is given a quadratic function f(x) = x^2 + ax + b and is asked to determine the coefficients 'a' and 'b', given that f(1) = 5 and f'(1) = 1.

Step 1: Use f(1) = 5 to find 'b'. Substitute 1 for x in the equation:

f(1) = 1 + a + b = 5

Step 2: To find 'a', use the derivative f'(x) = 2x + a and substitute x = 1 to get f'(1) = 1:

f'(1) = 2 + a = 1.

Solving both equations:

• f(1) = 1 + a + b = 5 implies that a + b = 4.
• f'(1) = 2 + a must equal 1, so a must be -1.
• With a = -1, substitute into a + b = 4 to find that b = 5.

Therefore, the coefficient b is 5, which corresponds to option D.