Determine the point at which the graph of the function has a horizontal tangent line.
f(x) = (9x^2)/(x^2+6)
To find the points at which the graph of the function has a horizontal tangent line, we need to find the values of x where the derivative of the function is zero.
First, let's find the derivative of the function f(x) = (9x^2)/(x^2+6):
f'(x) = [(18x)(x^2+6) - (9x^2)(2x)] / (x^2+6)^2
= (18x^3 + 108x - 18x^3) / (x^2+6)^2
= 108x / (x^2+6)^2
Next, we need to set the derivative equal to zero and solve for x:
108x / (x^2+6)^2 = 0
Since the numerator is zero (108x = 0), this equation will be satisfied when x = 0.
Therefore, the graph of the function f(x) = (9x^2)/(x^2+6) will have a horizontal tangent line at the point where x = 0.
To verify this, we could plot the function f(x) and find the point where the tangent line is horizontal, but in this case, we can see from the equation that the only point where the derivative is zero is x = 0.