Finding Local Max, Min, and POI of y=-x^5+5x-6

In summary: B+5x+-+6)&fp_xMin=-10&fp_xMax=10&fp_yMin=-10&fp_yMax=10&fp_xAxis=true&fp_yAxis=true&fp_width=600&fp_height=600&fp_animSpeed=1&fp_zoomX=1&fp_zoomY=1&fp_cursorCoords=true&fp_units=true&fp_hideGrid=false&fp_hideAxes=false&fp_labelAxes=true&fp_labelGrid=true&fp_showMouseCoords=true&fp_showSettings=true&fp_showHelp=true&fp_showInfo=true&fp_showFPS=true&fp_showTime=true&fp_autoplay=true&fp_anim
  • #1
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Homework Statement



a) Graph [tex]y = -x^5 +5x -6[/tex]. Include coordinates of local max, min, and points of inflection. Indicate the behavior as x-> -infinity, x -> infinity.

b) Find in slope-intercept form the equation of the tangent line to the curve at x = 2.

Homework Equations



The Attempt at a Solution



a) I found that the critical numbers are 1 and -1. The POI is 0. As x->infinity the limit -> -infinity and vice versa. Coordinates are (1,-2), (0,-6), (-1,-10). I've also graphed it to match my findings.

b) I'm not sure what to do for this part.

Thanks for any help!
 
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  • #2
duki said:

Homework Statement



a) Graph [tex]y = -x^5 +5x -6[/tex]. Include coordinates of local max, min, and points of inflection. Indicate the behavior as x-> -infinity, x -> infinity.

b) Find in slope-intercept form the equation of the tangent line to the curve at x = 2.

Homework Equations



The Attempt at a Solution



a) I found that the critical numbers are 1 and -1. The POI is 0. As x->infinity the limit -> -infinity and vice versa. Coordinates are (1,-2), (0,-6), (-1,-10). I've also graphed it to match my findings.

b) I'm not sure what to do for this part.

Part a) looks correct

For part b) you just need to find the equation of the tangent to the curve at that point. So you'll need the coordinate of y at x=2 and the gradient at x=2. {gradient function=[itex]\frac{dy}{dx}[/itex]
 
  • #3
hmm... I'm getting like 2,-13 and 2,-75? Don't think I understand what you mean...
 
  • #4
For part b) your just asked for the equation of a line that is tangent to the curve.

Well you need a point and a slope. They give you the point: it's when x=2. Finding y' gives you the slopes at every point (btw gradient is a fancy word for slope in this case), we need the slope at x=2 because that's the point that the line needs to be tangent.
 
  • #5
so like [tex] y = -75x + -28[/tex] ?
 
  • #6
point slope form is:

[tex]y-y_0=m(x-x_0)[/tex]

Where (x_0,y_0) is your point and m is your slope.
 
  • #7

1. How do I find the local max and min of a given function?

The local max and min of a function can be found by taking the derivative of the function and setting it equal to zero. Then, solving for the x-values will give the coordinates of the local max and min points.

2. What is a point of inflection (POI) and how do I find it?

A point of inflection is a point on a graph where the concavity changes from concave up to concave down, or vice versa. To find the POI, take the second derivative of the function and set it equal to zero. The x-value of the solution will be the coordinate of the POI.

3. Can a function have multiple local max or min points?

Yes, a function can have multiple local max or min points. This occurs when the derivative of the function is equal to zero at multiple points.

4. Is there a shortcut for finding local max, min, and POI?

Yes, there is a shortcut called the first derivative test. This involves taking the derivative of the function and evaluating it at the critical points (points where the derivative is equal to zero). If the derivative is positive at a critical point, then that point is a local min. If the derivative is negative, then the point is a local max. If the derivative is zero, then further analysis is needed to determine if it is a local max, min, or POI.

5. How do I graph the function and identify the local max, min, and POI?

To graph the function, plot a few points and connect them with a smooth curve. Then, use the methods mentioned above to find the local max, min, and POI. Plot these points on the graph and connect them with a dotted line to indicate that they are not part of the actual function. This will give a visual representation of the local extrema and POI of the function.

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