Find the critical numbers of the function. (enter your answers as a comma-separated list. if an answer does not exist, enter dne.) f(x) = 6 + 1 4 x − 1 2 x2

f(x) = 6 + 14x - 12x^2 (please use "^" to denote exponentiation)

To find the critical numbers, find the first derivative dy/dx, set it = to 0, and solve the resulting equation for x:

f '(x) = 0 + 14 - 24x = 0 => 24x = 14 => x = 14/24 = 7/12

There is just one critical number here, and it is x = 7/12. Find the corresponding y-value. At this point, the tangent line to the graph is horizontal.

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Abdulazeez10 Abdulazeez10 · Answer 2 · 2021-10-04T13:07:34+02:00

The critical number of the given function is [tex]\frac{7}{12}[/tex] OR 7/12

A number is critical if it makes the derivative of the expression equal 0.

Therefore, to find the critical numbers of the given function, we will first determine its derivative.

From the question, the given function is f(x) = 6 + 1 4 x − 1 2 x2.

This can be properly written as

[tex]f(x) = 6 + 1 4 x -1 2 x^{2}[/tex]

Then the derivative is

[tex]f'(x) = 0+1 4 -24 x[/tex]

∴ [tex]f'(x) = 1 4 -24 x[/tex]

Hence, the derivative of the given function is 14 - 24x

Now, to determine the critical numbers, we will set the derivative equals to 0.

That is,

[tex]14 - 24x = 0[/tex]

Then,

[tex]14 = 24x[/tex]

Now, divide both sides by 24

[tex]\frac{14}{24} =\frac{24x}{24}[/tex]

[tex]\frac{7}{12} = x[/tex]

∴ [tex]x =\frac{7}{12}[/tex]

Hence, the critical number of the given function is [tex]\frac{7}{12}[/tex] OR 7/12

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