## anonymous one year ago The figure below shows the graph of f ′, the derivative of the function f, on the closed interval from x = −2 to x = 6. The graph of the derivative has horizontal tangent lines at x = 2 and x = 4. Find the x-value where f attains its absolute maximum value on the closed interval from x = −2 to x = 6. Justify your answer

1. anonymous

2. anonymous

@Empty @IrishBoy123

3. ganeshie8

|dw:1439314511563:dw|

4. anonymous

|dw:1439314520378:dw|

5. ganeshie8

what do you know about the relationship between first derivative and relative min/max ?

6. anonymous

wherever the first derivative = zero their is likely a max min , where the first derivative is negative and then positive the function is concave down and the inverse of that to indicate concave up

7. anonymous

umm thats pretty much it, am i missing something

8. ganeshie8

that looks good, look at the given graph of first derivative notice that the first derivative stays negative in the interval [-2, 5], this means the function is "decreasing" in this interval, yes ?

9. anonymous

yep

10. ganeshie8

we cannot have any max/min in the interval (-2, 5) since the function is continuously decreasing

11. ganeshie8

what about the point (5, 0) does that mean the function has a min or max at x=5 ?

12. anonymous

min

13. ganeshie8

good, since the first derivative is going form "negative" to "positive", the function will have a local minimum at x=5

14. ganeshie8

that essentially means, we do not have any local maximums in the interval (-2, 6)

15. ganeshie8

so the absolute maximum must occur at the boundary points

16. anonymous

x = 6!!

17. ganeshie8

how do you know ?

18. anonymous

well it just kept increasing after x = 4 so i figured it would be the highest , but i guess x=-2 could be just as high because we don't know what happened before x = -2 right?

19. ganeshie8

Notice that the first derivative is "negative" in the interval (-2, 5) that means the actual function is "decreasing" in the interval (-2, 5)

20. anonymous

okay i see so your suggesting that because it decreased for such a long interval and then increased for such a short interval the maximum would be ant x = -2 is that what your saying?

21. ganeshie8

Kindof! but thats not it, at this point, i do believe the function attains its maximum at x = -2

22. anonymous

okay well why do you suggest its x = -2?

23. ganeshie8

|dw:1439315773571:dw|

24. ganeshie8

which area do you think is more black or red ?

25. anonymous

well grey obviously

26. ganeshie8

That means the actual function did not recover the fall yet

27. ganeshie8

so x=-2 is indeed the absolute maximum

28. anonymous

okay that makes sense

29. ganeshie8

If you're good with definite integrals, notice below : $f(6)-f(-2) = \int\limits_{-2}^6f'(x)\,dx \lt 0$ that implies $$f(6)\lt f(-2)$$

30. anonymous

okay thanks for the help!

31. ganeshie8

yw

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