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?

Answer 1

at #(1,2)#; equation of #y=2#

A horizontal tangent occurs whenever the function's derivative equals #0#, since a value of #0# represents that the function's tangent line has a slope of #0#. Lines with slope #0# are horizontal.

To find the function's derivative, use the power rule.

#f(x)=x^4-4x+5#
#f'(x)=4x^3-4#
Find the points when #f'(x)=0#.
#4x^3-4=0#
#4x^3=4#
#x^3=1#
#x=1#
There is a horizontal tangent at #(1,2)#, thus its equation is #y=2#.
We can check a graph of #f(x)#:

graph{(x^4-4x+5-y)(y-0x-2)=0 [-19.92, 20.63, -3.52, 16.74]}

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Answer 2

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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Answer from HIX Tutor

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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