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)=x-x^(2/3)(5/2-x)#?

Answer 1

See explanation and graph.

Rearranging,

#y=f(x)=(x^(5/3)+x-2.5x^(2/3))#

The graph passes through the origin (0, 0)

x-intercept ( y = 0 ): 1.4, nearly.

As #x to +-oo, y to +-oo#.

So, there are no asymptotes.

#y'=5/3x^(2/3)+1-(5/3)/x^(1/3)#
At x =0, y' has infinite discontinuity. It changes from #oo# to# -oo#.
and becomes 0 near x = 0.5, for sign change, from # -# to +
For #x < 0. y uarr, x in (0, 0.5), y darr, and x > 0,5# ( nearly), #y uarr#,

again .

There is a turning point near x = 0.5, for

the local minimum #= y(0.5)= -0.76#, nearly.,

,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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Answer 2

To sketch the graph of ( f(x) = x - x^{\frac{2}{3}}(5 - \frac{x}{2}) ), follow these steps:

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

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

  3. 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.

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

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

  6. 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.

  7. 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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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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