Given f(x)=x^3−2x+5/x+4 and f′(3)=a/b, where a and b are coprime positive integers, what is the value of a+b?
Details and assumptions
f′(x) denotes the derivative of f(x).
Is your question is f(x)= x^3-2x+(5/x)+4 ?
Xcube -2x +5 divide by x+4
is it x^3 -(2x+5)/(x+4) ?
To find the value of a+b, we first need to find the derivative of the given function, f(x).
To find the derivative, f'(x), we can use the quotient rule. The quotient rule states that if we have a function of the form f(x) = g(x)/h(x), where g(x) and h(x) are both differentiable functions, then the derivative is given by:
f'(x) = (g'(x) * h(x) - g(x) * h'(x)) / (h(x))^2
In this case, g(x) = x^3 - 2x + 5 and h(x) = x + 4. Let's find the derivatives of g(x) and h(x):
g'(x) = 3x^2 - 2
h'(x) = 1
Now, we can substitute the values into the quotient rule formula:
f'(x) = [(3x^2 - 2) * (x + 4) - (x^3 - 2x + 5) * 1] / (x + 4)^2
Next, we need to find the derivative at x = 3, which is denoted as f'(3). To find this, we substitute x = 3 into the derivative equation:
f'(3) = [(3(3)^2 - 2) * (3 + 4) - (3^3 - 2(3) + 5) * 1] / (3 + 4)^2
Now, let's simplify the equation:
f'(3) = [(27 - 2) * 7 - (27 - 6 + 5) * 1] / (7)^2
= [25 * 7 - 26 * 1] / 49
= (175 - 26) / 49
= 149 / 49
Since a and b are coprime positive integers, we need to simplify the fraction 149/49. The greatest common divisor (GCD) of 149 and 49 is 1, so the fraction is already in its simplest form.
Therefore, a = 149 and b = 49. The value of a+b is 149 + 49 = 198.