How do you find the points where the graph of the function #f(x) = x^4 - 4x^3# has horizontal tangents?

Answer 1

That will happen when the derivate is zero. In x=0 and x=3.

#(x^4-4x^3)'=4x^3-12x^2#
#4x^3-12x^2=4x^2(x-3)#, which as zeros in x=0 and x=3.
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Answer 2

To find the points where the graph of the function f(x) = x^4 - 4x^3 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 - 12x^2.

Next, we set the derivative equal to zero and solve for x: 4x^3 - 12x^2 = 0.

Factoring out 4x^2, we get: 4x^2(x - 3) = 0.

Setting each factor equal to zero, we have two possibilities: 4x^2 = 0 or x - 3 = 0.

Solving 4x^2 = 0, we find x = 0.

Solving x - 3 = 0, we find x = 3.

Therefore, the points where the graph of f(x) = x^4 - 4x^3 has horizontal tangents are (0, f(0)) and (3, f(3)).

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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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