How do you find the points where the graph of the function #f(x) = x^4-4x+5# has horizontal tangents and what is the equation?
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To find the function's derivative, use the power rule.
graph{(x^4-4x+5-y)(y-0x-2)=0 [-19.92, 20.63, -3.52, 16.74]}
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To find the points where the graph of the function f(x) = x^4-4x+5 has horizontal tangents, we need to find the values of x where the derivative of the function is equal to zero.
First, we find the derivative of f(x) by applying the power rule: f'(x) = 4x^3 - 4.
Next, we set the derivative equal to zero and solve for x: 4x^3 - 4 = 0.
Simplifying the equation, we have: x^3 - 1 = 0.
Factoring the equation, we get: (x - 1)(x^2 + x + 1) = 0.
Setting each factor equal to zero, we find two possible values for x: x = 1 and x^2 + x + 1 = 0.
Solving the quadratic equation x^2 + x + 1 = 0, we find that it has no real solutions.
Therefore, the only point where the graph of f(x) has a horizontal tangent is when x = 1.
The equation of the horizontal tangent line at x = 1 can be found by substituting x = 1 into the original function f(x): f(1) = 1^4 - 4(1) + 5 = 2.
Thus, the equation of the horizontal tangent line is y = 2.
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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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