The function r(w) = 2aw-2 + b + c where a, b and c are constants.



You are given that r(1) = 5; r’(4) = 12 and r′’(2) = 18.
By substituting for w in the above function and its derivatives or otherwise, show that the three
given function values give rise to a system of linear equations in the unknowns a, b and c

To show that the given function values give rise to a system of linear equations in the unknowns a, b, and c, we can substitute the values of w into the function and its derivatives.

1. Start with the given function: r(w) = 2aw-2 + b + c

Substitute w = 1: r(1) = 2a(1)-2 + b + c = 5

This gives us the equation: 2a - 2 + b + c = 5

2. Take the derivative of the function: r'(w) = d(r)/dw = d(2aw-2 + b + c)/dw = 2a

Substitute w = 4: r'(4) = 2a = 12

This gives us the equation: 2a = 12

3. Take the second derivative of the function: r''(w) = d²(r)/dw² = d²(2aw-2 + b + c)/dw² = 0

Substitute w = 2: r''(2) = 0 = 18

This gives us the equation: 0 = 18

Now let's summarize the equations we obtained:

Equation 1: 2a - 2 + b + c = 5
Equation 2: 2a = 12
Equation 3: 0 = 18

As you can see, these equations form a system of linear equations in the unknowns a, b, and c.