At what values of x does the function f(x)= x(x-4)^4 have a horizontal tangent line?
To find the values of x at which the function f(x) = x(x-4)^4 has a horizontal tangent line, we need to find the critical points of the function. A critical point occurs when the derivative of the function is equal to zero or undefined.
Let's first find the derivative of the function f(x) = x(x-4)^4. The derivative of a function can be found using the product rule and the chain rule:
f'(x) = [x * d/dx[(x-4)^4]] + [(x-4)^4 * d/dx[x]]
Now, let's calculate the derivatives:
d/dx[(x-4)^4] = 4(x-4)^3 * d/dx[(x-4)]
= 4(x-4)^3 * (1)
= 4(x-4)^3
d/dx[x] = 1
Plugging these derivatives back into the equation:
f'(x) = [x * 4(x-4)^3] + [(x-4)^4 * 1]
= 4x(x-4)^3 + (x-4)^4
= (x-4)^3[4x + (x-4)]
= (x-4)^3(5x-16)
To find the values of x where the derivative is equal to zero, we set f'(x) = 0:
(x-4)^3(5x-16) = 0
The derivative is equal to zero if either (x-4)^3 = 0 or 5x-16 = 0.
For (x-4)^3 = 0, we solve for x:
(x-4)^3 = 0
x - 4 = 0
x = 4
For 5x-16 = 0, we solve for x:
5x - 16 = 0
5x = 16
x = 16/5
Therefore, the function f(x) = x(x-4)^4 has a horizontal tangent line at x = 4 and x = 16/5.