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anonymous
 one year ago
The graph of f ′ (x), the derivative of f of x, is continuous for all x and consists of five line segments as shown below. Given f (0) = 7, find the absolute minimum value of f (x) over the interval [–3, 0].
anonymous
 one year ago
The graph of f ′ (x), the derivative of f of x, is continuous for all x and consists of five line segments as shown below. Given f (0) = 7, find the absolute minimum value of f (x) over the interval [–3, 0].

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anonymous
 one year ago
Best ResponseYou've already chosen the best response.0First, I'm not sure but if maybe this makes sense to you, it's worth a try Ok to find the minimum value of f(x) in the interval [3,0] we have to evaluate f(x) at the points 3, 0 and we also have to find the value x for which f'(x)=0 because the value of x for which f'(x)=0 is the value of x at which the function f(x) has a critical point(it's either maximum or minimum) Now from the graph we know f'(x)=0 for x=0 therefore at the value x=0 we have \[f(0)=7\] Since x=0 is also the end point in the interval, that's 2 out of 3 points done Now we need to look at x=3 Consider the definite integral \[\int\limits_{3}^{0}f'(x)dx=f(0)f(3)\]\[\implies f(3)=f(0)\int\limits_{3}^{0}f'(x)dx\] Using the graph you can find the f'(x) in the interval 3 to 0 as an equation of a straight line

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0okay sorry had to go to the restroom here are the answer choices by the way 0 2.5 4.5 11.5

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0okay so if the equation is a straight line then their is no minimum right ?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Since x=0 is the value of x for which f'(x)=0, f(0) gives the value of f(x) at which it is either maximum or minimum thus the value 7 is either it's maximum or minimum now to check whether it's maximum or minimum we'll have to look at how f'(x) changes value when it passes through 0 If f'(x) changes from negative to positive ie, f'(x)>0 for some x<c and f'(x)<0 for some x>c, then it's a point of local maxima here the c is 0, it's the value of x at which f'(x) is 0 so take any value less than 0 and more than 0 and look at f'(x) and if it's f'(x)<0 and f'(x)>0 for x<c and x>c respectively then it's minima

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0umm wow okay do you mean positive and negative with respect the the x or the y axis?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0take value of x less than 0, look at the corresponding value of f'(x) if it's less than 0 and take value of x more than 0 look at the corresponding value of f'(x) if it's more than 0 it's a local minima and the value of 7 would be minimum if that's the case otherwise maximum

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0and if it does not change sign it's not a maxima or minima

phi
 one year ago
Best ResponseYou've already chosen the best response.0The integral of f'(x) is the area under the curve. if we had the value at f(4), then we could find f(x) for any x from 4 to +5 by adding the area from 4 up to the x of interest to f(4). in this case we have f(0), so we subtract the area under the curve between x=3 and x=0 from f(0)

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0give me a second to catch my breath

phi
 one year ago
Best ResponseYou've already chosen the best response.0hopefully it is clear that starting at f(4) the area under the curve is always going up as we move to the right... f(x) is constantly going up. if we want the min in the range of 3 to 0 then we want f(3) ...

phi
 one year ago
Best ResponseYou've already chosen the best response.0we can say f(3) + area under the curve from 3 to 0 = f(0) = 7 thus f(3)= 7  area of triangle

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0if the area is increasing and then starts to decrease after x = 3 wouldn't that make x = 3 a maximum ?

mathmate
 one year ago
Best ResponseYou've already chosen the best response.0dw:1439293299504:dw

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Yep it seems like the point f(0) is not of our use, we must evaluate f(3) use the equation \[f'(x)=x\] and integrate it within the limits 3,0 because f'(x) is not changing sign for a value of x less 0 and more than 0 (it's positive in both case) so it's not a point of min or max the equation I gave is the equation of the 2nd line segment from left in the graph

phi
 one year ago
Best ResponseYou've already chosen the best response.0the area of the triangle has base 3 and height 3 , so area 4.5 and f(3)= 74.5= 2.5

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0where did you get f'(x) = x

phi
 one year ago
Best ResponseYou've already chosen the best response.0***if the area is increasing and then starts to decrease after x = 3**** the area does not decrease... as you move to the left you have more area (imagine painting the region between the curve and the xaxis) it is true the area increases at a slower rate, but it still goes up. for the area to decrease the curve must go *below the xaxis* (that counts as negative area)

phi
 one year ago
Best ResponseYou've already chosen the best response.0**as you move from left to right

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0\[y=(\frac{y_{2}y_{1}}{x_{2}x_{1}})x+C\] Equation of a straight line if the line passes through origin, c=0 \[y=f'(x)=(\frac{y_{2}y_{1}}{x_{2}x_{1}})x\] take any pair of points lying in the line segment now \[f'(x)=\frac{21}{2(1)}.x=\frac{1}{1}.x=x\]

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0with this you can find f(3) \[f(3)=f(0)\int\limits_{3}^{0}f'(x)dx=7\int\limits_{3}^{0}x.dx=7+\int\limits_{3}^{0}x.dx\]

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0compare the value of \[f(0)\] and \[f(3)\] you can tell which is less, whichever value is less is the minimum value forget some of the earlier stuff I said about looking at the sign, that's not required here

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0well obviously f(3)

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0okay i think i'm still unsure how to do this ,iv got another problem similar to this one could you help me with that one too?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0did you understand how I got f'(x)=x
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