Find the all the possible values for x if there are any so that the graph of the function has a horizontal tangent.
y(x)=x^4-4x+4
just recall that a horizontal line has slope=0. So, you want
y' = 4x^3-4 = 0
x=1
To find the values of x for which the graph of the function has a horizontal tangent, we need to find the derivative of the function and then solve for x when the derivative equals zero.
Let's start by finding the derivative of the function y(x) = x^4 - 4x + 4.
To find the derivative, we take the derivative of each term separately using the power rule. The power rule states that if we have a term of the form ax^n, the derivative with respect to x is given by nx^(n-1).
Taking the derivative of the first term x^4, we get: 4x^3.
Taking the derivative of the second term -4x, we get: -4.
Since the derivative of a constant term (like 4) is always zero, it doesn't affect the horizontal tangent.
So, the derivative of the function y(x) = x^4 - 4x + 4 is: y'(x) = 4x^3 - 4.
Now, we set this derivative equal to zero to find the values of x where there is a horizontal tangent:
4x^3 - 4 = 0
Adding 4 to both sides, we get:
4x^3 = 4
Dividing both sides by 4, we have:
x^3 = 1
Now, we need to find the values of x that satisfy this equation.
One solution that satisfies x^3 = 1 is x = 1, because 1^3 = 1.
However, there might be more solutions, because x^3 = 1 also has complex roots.
To find all the solutions, we can use the fact that x^3 = 1 can be factored as (x - 1)(x^2 + x + 1) = 0.
Setting each factor equal to zero, we get two additional solutions:
x - 1 = 0 => x = 1
x^2 + x + 1 = 0
To solve this quadratic equation, we can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
In this case, a = 1, b = 1, and c = 1. Substituting these values into the quadratic formula:
x = (-(1) ± √((1)^2 - 4(1)(1))) / (2(1))
Simplifying the expression further:
x = (-1 ± √(1 - 4)) / 2
x = (-1 ± √(-3)) / 2
Since the square root of a negative number is not a real number, the quadratic equation x^2 + x + 1 = 0 does not have real solutions.
Therefore, the only value of x for which the graph of the function y(x) = x^4 - 4x + 4 has a horizontal tangent is x = 1.