# How do you find the inflection points of the graph of the function: # f(x) = (6x)/(x^2 + 16)#?

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To find the inflection points of the graph of the function ( f(x) = \frac{6x}{x^2 + 16} ), follow these steps:

- Find the second derivative of the function ( f(x) ).
- Set the second derivative equal to zero and solve for ( x ).
- Determine whether the solutions obtained in step 2 correspond to inflection points by analyzing the concavity of the graph.

Let's begin with step 1:

First derivative: [ f'(x) = \frac{d}{dx} \left( \frac{6x}{x^2 + 16} \right) ]

Using the quotient rule: [ f'(x) = \frac{(6)(x^2 + 16) - (6x)(2x)}{(x^2 + 16)^2} ]

Simplify: [ f'(x) = \frac{6x^2 + 96 - 12x^2}{(x^2 + 16)^2} ] [ f'(x) = \frac{-6x^2 + 96}{(x^2 + 16)^2} ]

Second derivative: [ f''(x) = \frac{d}{dx} \left( \frac{-6x^2 + 96}{(x^2 + 16)^2} \right) ]

Using the quotient rule: [ f''(x) = \frac{(x^2 + 16)^2 \cdot (-12x) - (-6x^2 + 96) \cdot 2(x^2 + 16)(2x)}{(x^2 + 16)^4} ]

Simplify: [ f''(x) = \frac{-12x(x^2 + 16)^2 + 24x(-6x^2 + 96)(x^2 + 16)}{(x^2 + 16)^4} ] [ f''(x) = \frac{-12x(x^2 + 16) + 24x(-6x^2 + 96)}{(x^2 + 16)^3} ]

[ f''(x) = \frac{-12x^3 - 192x + 144x^3 + 2304x}{(x^2 + 16)^3} ] [ f''(x) = \frac{132x^3 + 2112x}{(x^2 + 16)^3} ]

Now, move on to step 2:

Set ( f''(x) = 0 ): [ \frac{132x^3 + 2112x}{(x^2 + 16)^3} = 0 ]

Since the numerator cannot be zero, set ( 132x^3 + 2112x = 0 ): [ 132x(x^2 + 16) = 0 ]

This gives us two solutions: ( x = 0 ) and ( x^2 + 16 = 0 ), which has no real solutions.

Finally, for step 3, analyze the concavity of the graph around the critical points:

Evaluate the sign of ( f''(x) ) around the critical point ( x = 0 ) to determine the concavity: [ f''(x) = \frac{132x^3 + 2112x}{(x^2 + 16)^3} ]

Substitute a value from each interval into ( f''(x) ) (e.g., ( x = -1 ) and ( x = 1 )): [ f''(-1) = \frac{132(-1)^3 + 2112(-1)}{((-1)^2 + 16)^3} ] [ f''(1) = \frac{132(1)^3 + 2112(1)}{(1^2 + 16)^3} ]

After calculating, you'll find that ( f''(-1) ) and ( f''(1) ) have opposite signs, indicating a change in concavity around ( x = 0 ). Hence, ( x = 0 ) is an inflection point.

Therefore, the inflection point of the graph of the function ( f(x) = \frac{6x}{x^2 + 16} ) is ( x = 0 ).

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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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