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?

Answer 1

Points are #x=4.045# and #x=-0.934# and tangents are #y=178.955# and #y=364.0999#

Slope of a tangent is given by #(df(x))/(dx)# and tangent is horizontal when slope is #0#.
In otherwords, one will find horizontal tangents, where #(df)/(dx)=0#
As #f(x)=3x^3-14x^2-34x+347#, we have
#(df)/(dx)=9x^2-28x-34#
and #9x^2-28x-34=0#
i.e. #x=(28+-sqrt(28^2-4*9*(-34)))/18#
= #(28+-sqrt(784+1224))/18#
= #(28+-44.81)/18#
= #72.81/18# or #-16.81/18#
= #4.045# or #-0.934#
Further when #x=4.045#, #f(x)=178.955#
and when #x=-0.934#, #f(x)=364.099#
Hence tangents are #y=178.955# and #y=364.0999#
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Answer 2

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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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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