# How do you find the points where the graph of the function #(3x^3) - (14x^2) -(34x) + 347# has horizontal tangents and what is the equation?

Points are

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To find the points where the graph of the function (3x^3) - (14x^2) - (34x) + 347 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 the function by taking the derivative of each term separately:

f'(x) = 9x^2 - 28x - 34

Next, we set the derivative equal to zero and solve for x:

9x^2 - 28x - 34 = 0

Using the quadratic formula, we find the solutions for x:

x = (-(-28) ± √((-28)^2 - 4(9)(-34))) / (2(9))

Simplifying the equation, we get:

x = (28 ± √(784 + 1224)) / 18

x = (28 ± √2008) / 18

Finally, we simplify the solutions:

x = (28 ± 2√502) / 18

Therefore, the points where the graph of the function has horizontal tangents are given by the x-values:

x = (28 + 2√502) / 18

x = (28 - 2√502) / 18

The corresponding equations of the horizontal tangents can be found by substituting these x-values back into the original function:

y = (3((28 + 2√502) / 18)^3) - (14((28 + 2√502) / 18)^2) - (34((28 + 2√502) / 18)) + 347

y = (3((28 - 2√502) / 18)^3) - (14((28 - 2√502) / 18)^2) - (34((28 - 2√502) / 18)) + 347

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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.

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