.
The gradient/slope of a function of f(x) = (2x-b)^a is f'(x) = 24x^2-24x+6,
find the constant a and b. I'm a confused on how to start on this.
~~~~~~~~~~~~~~~~~~~
The wording of the problem means that the derivative of the function f(x) = (2x-b)^a
is the polynomial of degree 2, f'(x) = 24x^2 - 24x + 6.
It means that f(x) is a polynomial of degree 3; hence, a= 3.
Next, you can make FOIL of (2x-b)^3 to get (2x-b)^3 = 8x^3 - 3*(2x^2)*b + . . . = 8x^2 - 12x^2*b + . . .
Then, looking at its term of degree 2, -3*(2x^2)*b = -3*(4x^2)*b = -12x^2*b, and comparing it
with its derivative -24x, you conclude that b= 1.
So, the constants "a" and "b" are a= 3, b= 1.
Thus f(x) = (2x-1)^3 = 8x^3 - 3*(2x)^2*1 + 3*(2x)*1^2 - 1 = 8x^3 - 12x^2 + 6x - 1.
ANSWER. a= 3, b= 1.
Solved.