How do you sketch the graph by determining all relative max and min, inflection points, finding intervals of increasing, decreasing and any asymptotes given #f(x)=(4x)/(x^2+1)#?
#f(x)# starts from#y=0# and keeps decreasing until#x=-1# where it has a minimum for#y=-2#
#f(x)# increases for#x in (-1,1)# reaching a maximum for#x=1# at#y=2#
Finally, for#x>1# #f(x)# is strictly decreasing and approaches#y=0# from positive values.
Now we can determine:
We can conclude that:
graph{(4x)/(x^2+1) [-10, 10, -5, 5]}
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To sketch the graph of ( f(x) = \frac{4x}{x^2 + 1} ), follow these steps:
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Find the first derivative of ( f(x) ) to determine critical points. ( f'(x) = \frac{d}{dx}\left(\frac{4x}{x^2 + 1}\right) )
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Set ( f'(x) = 0 ) and solve for ( x ) to find critical points.
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Determine the second derivative ( f''(x) ) to classify critical points as relative maxima, minima, or inflection points.
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Find the intervals of increasing and decreasing by analyzing the sign of ( f'(x) ) between critical points.
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Identify any asymptotes by examining the behavior of ( f(x) ) as ( x ) approaches positive or negative infinity.
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Sketch the graph incorporating all the information gathered.
Let's proceed with these steps:
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Find the first derivative: ( f'(x) = \frac{d}{dx}\left(\frac{4x}{x^2 + 1}\right) = \frac{(4(x^2 + 1) - 4x(2x))}{(x^2 + 1)^2} )
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Set ( f'(x) = 0 ) and solve for ( x ) to find critical points: ( 4(x^2 + 1) - 4x(2x) = 0 ) ( 4x^2 + 4 - 8x^2 = 0 ) ( -4x^2 + 4 = 0 ) ( x^2 = 1 ) ( x = \pm 1 )
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Determine the second derivative: ( f''(x) = \frac{d}{dx}\left(\frac{(4(x^2 + 1) - 4x(2x))}{(x^2 + 1)^2}\right) )
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( f''(x) = \frac{(4(2x) - 4(2x) - 8x^2) \cdot (x^2 + 1)^2 - 2(2x)(4(x^2 + 1) - 4x(2x)) \cdot (2(x^2 + 1) \cdot (2x))}{(x^2 + 1)^4} ) Simplifying further would be a bit cumbersome here.
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Determine intervals of increasing and decreasing: Test points within intervals determined by critical points and analyze the sign of ( f'(x) ).
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Identify asymptotes: Check the behavior of ( f(x) ) as ( x ) approaches positive or negative infinity.
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Sketch the graph based on the above information.
Please note that for a more precise graph, you may want to use a graphing calculator or software to plot the function.
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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.
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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- What are the points of inflection of #f(x)= 3sin2x - 2xcosx# on #x in [0, 2pi] #?
- If #1/3(x-1)^3+2#, what are the points of inflection, concavity and critical points?
- How do you use the first and second derivatives to sketch #y=e^x/x#?

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