Question

PLEASE HELP suppose that g is a function with the following two properties: g(-x)=g(x) for all x, and gPRIME(a) exists. which of the following must necessarily be equal to gPRIME(-a).
1) gPRIME(a)
2) -gPRIME(a)
3) (1)/(gPRIME(a))
4)-(1)/(gPRIME(a))
5) none
an explanation would be greatly appreciated.

Answer 1

g(-x)=g(x)
means the left side (of zero) looks like the right side
a simple example

Answer 2

g'(a) means the derivative of g(x) at x=a
in other words, the slope (of the tangent line to the curve)

in the simple example up above, the derivative is the slope of the line
can you see the relationship between

slope at x=a and the slope "on the other side of zero" at x= -a ?

Answer 3

don't they have similar slopes or something? @phi

Answer 4

like reciprocals? i'm not sure

Answer 5

yes, clear the slopes are related, but you can see on the right side the lines goes up
and on the left side the line goes down

say the line on the right has a slope of 2
every time you move 1 over in the x direction, you move 2 up in the y
the other side is the same shape so it must be every time you move 1 to the right, you move 2 in the y direction. but you can see it is down
so its slope is -2

Answer 6

i see, they are the same but one is negative and the other positive

Answer 7

yes. and the idea works for any (complicated) shape
that has g(x)= g(-x)