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)=xx^(2/3)(5/2x)#?
See explanation and graph.
Rearranging,
The graph passes through the origin (0, 0)
xintercept ( y = 0 ): 1.4, nearly.
So, there are no asymptotes.
again .
There is a turning point near x = 0.5, for
,Relative maximum y = 0, at the cusp (0, 0).
The origin is a cusp and the tangent does not cross the curve,
there is no point of inflexion
graph{x^(2/3)(2.5+x^(1/3)+x) [2.5, 2.5, 1.25, 1.25]}
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To sketch the graph of ( f(x) = x  x^{\frac{2}{3}}(5  \frac{x}{2}) ), follow these steps:

Find the Derivative: ( f'(x) = 1  \frac{2}{3}x^{\frac{1}{3}}(5  \frac{x}{2})  x^{\frac{2}{3}}(\frac{1}{2}) ).

Solve for Critical Points: Set ( f'(x) = 0 ) and solve for ( x ). This gives critical points.

Determine the Nature of Critical Points: Use the first derivative test or the second derivative test to determine if each critical point is a relative maximum, relative minimum, or neither.

Find Inflection Points: Find where the second derivative changes sign. This indicates inflection points.

Determine Intervals of Increase and Decrease: Use the sign of the first derivative to determine where the function is increasing or decreasing.

Find Vertical Asymptotes: Determine if there are any vertical asymptotes by finding where the function approaches infinity or negative infinity as ( x ) approaches certain values.

Find Horizontal Asymptotes: Determine if there are any horizontal asymptotes by finding the limit of the function as ( x ) approaches positive or negative infinity.
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When evaluating a onesided 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 onesided 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 onesided 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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