Question

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

Answer 1

@Empty @IrishBoy123

Answer 2

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

Answer 3

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

Answer 4

umm thats pretty much it, am i missing something

Answer 5

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 ?

Answer 6

yep

Answer 7

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

Answer 8

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

Answer 9

min

Answer 10

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

Answer 11

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

Answer 12

so the absolute maximum must occur at the boundary points

Answer 13

x = 6!!

Answer 14

how do you know ?

Answer 15

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?

Answer 16

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

Answer 17

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?

Answer 18

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

Answer 19

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