Can someone please show steps as I am having a hard time understanding. The position (in meters) of a particle is modeled by y=x^3-12.5x^2+51x-67.5, where xis in seconds. Find the max, min and point(s) of inflection.
5 answers
Reiny gave you some tips. Where do you get stuck?
we got a few examples in class but not understanding it at all
Oh, please. There must be something you understand by this time in the course.
You know that max/min occurs where y'=0. Do you know how to find y'?
You know that max/min occurs where y'=0. Do you know how to find y'?
yes, i pretty sure i got it as my answer mathes the back of the book
Well, here goes. If you really are totally lost, you'll probably have much better progress talking to someone in person.
y=x^3-12.5x^2+51x-67.5
Now, you know that y' is the slope of the tangent line to the curve at (x,y). When that tangent line is horizontal, the curve is either at a max or a min. Think of a hill or valley -- at the top of the hill, you stop rising (slope >0), and start falling (slope <0)
y' = 3x^2 -25x + 51
to find y'=0, solve for x
0 = 3x^2 - 25x + 51
x = 1/6 (25 ±√13)
x = 3.57 or 4.77
Now, from your (presumed) general knowledge of the shape of cubics, you should be able to surmise that the first value is where y reaches a max, the second where it reaches a min.
A point of inflection occurs when the slope reaches a max or min. That's where the curve changes from concave up to concave down. It is where y''=0 and y' ≠ 0.
y'' = 6x - 25
y'' = 0 when x = 25/6 = 4.17
plug in those x-values to find the actual y-coordinates of the points.
y=x^3-12.5x^2+51x-67.5
Now, you know that y' is the slope of the tangent line to the curve at (x,y). When that tangent line is horizontal, the curve is either at a max or a min. Think of a hill or valley -- at the top of the hill, you stop rising (slope >0), and start falling (slope <0)
y' = 3x^2 -25x + 51
to find y'=0, solve for x
0 = 3x^2 - 25x + 51
x = 1/6 (25 ±√13)
x = 3.57 or 4.77
Now, from your (presumed) general knowledge of the shape of cubics, you should be able to surmise that the first value is where y reaches a max, the second where it reaches a min.
A point of inflection occurs when the slope reaches a max or min. That's where the curve changes from concave up to concave down. It is where y''=0 and y' ≠ 0.
y'' = 6x - 25
y'' = 0 when x = 25/6 = 4.17
plug in those x-values to find the actual y-coordinates of the points.
2 answers