Below is the graph of f ′(x), the derivative of f(x), and has x-intercepts at x = –3, x = 1 and x = 2. there are horizontal tangents at x = –1.5 and x = 1.5. which of the following statements is true? a) f has a relative minimum at x = 1.5 b) f has relative maximum at x = -1.5 c) f is decreasing on the interval from x = 1 to x = 2 d) none of these are true

the derivitive is just the slope

minimum happens when the derivitive goes from negative to positive, imagine a slope of the function, the minimum is where the slope goes from neative to positive, and to get there, it has to pass through 0

max happens when the derivitive goes from positive to negative

increaseing is when the derivitive is positive

so, based on what you said, the slope of f(x) is 0 at x=-3, x=1 and x=2 since those are where the derivitive is 0 (derivitive is just the slope)

A and B are wrong because the derivitive isn't 0 at those points

C is correct because increasing means that the derivitive is positive, and so therefo since the only hoirontal place in between 1 and 2 is 1.5, it must remain positive throughout and not dip down, C is right

D is wrong then

answer is C

Part A:

5x+3(x−5)=6x+2x−15

Distribute

5x+3x + 3(−5) = 6x + 2x − 15

5x + 3x + −15 = 6x + 2x − 15

Combine Like Terms

(5x + 3x) − 15 = 6x + 2x + −15  

8x + −15 = 8x + −15  

8x − 15 =8x − 15

Subtract 15 from both sides.

8x = 8x

There are infinite  answers.

Part B:

I used the distributive property.