Question

Let f be a differentiable function with the following properties:

i. f(x) = ax² + bx
ii. f(1) = 6 and f'(x) = 6
iii. ∫(from 1 to 2) f(x) dx = 14

What is the value of a?

a) 2
b) 3
c) 4
d) 5

Answer 1

To find the value of a, we can use the given information and set up and solve equations involving the properties of the function. Using the derivative of the function and the integral property, we can determine the value of a.

Step-by-step explanation:

To find the value of a, we can use the given information to set up and solve equations.

1. From property i, f(x) = ax² + bx.
2. From property ii, f'(x) = 6.
3. Using property i and ii, we can find f'(x) by taking the derivative of f(x). In this case, f'(x) = 2ax + b = 6.
4. From property iii, ∫(from 1 to 2) f(x) dx = 14.
5. Using the antiderivative of ax² + bx, we have ∫(from 1 to 2) (ax² + bx) dx = 14. Upon solving this, we can find the value of a.

By calculating the antiderivative and evaluating it at the limits of integration, we get (2a/3)(2³ - 1³) + (b/2)(2² - 1²) = 14. Simplifying this equation, we can find the value of a.