Given the graph of f ′(x), the derivative of f(x), which of the following statements are true about the graph of f(x)?

Graph is parabola with x intercepts at x equals 0 and 3 and vertex at x equals 1.

The graph of f has a critical point at x = 0.
The graph of f has a critical point at x = 1.
The graph of f has a point of inflection at x = 1.
I only
II only
III only
I and III only

this is the derivitive of x
critical points are where the first derivitive is equal to 0 or where the derivitive doesn't exist

ok, it is 0 at x=0 and 2
so I is true but not II

the point of inflection is where the slope of the 1st derivitive=0
that is at x=1
so III is true

I and III

4th option

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deepakrai138 deepakrai138 · Answer 2 · 2021-12-27T06:49:57+01:00

Correct option is statement I and III only.

Given Graph is parabola with x intercepts at x equals 0 and 3 and vertex at x equals 1.

1 statement -The graph of f has a critical point at [tex]x = 0[/tex].

Since critical points occur when [tex]f'(x) = 0[/tex] or [tex]f'(x) =[/tex] Does not exist, at [tex]x = 0[/tex] on your graph above [tex]f'(x) = 0[/tex] so this is a critical point and the statement is True.

2. statement- The graph of f has a critical point at [tex]x = 1.[/tex]

Since[tex]f'(x)[/tex] is not 0 at [tex]x = 1[/tex] and it does exist at that point, [tex]x = 1[/tex] is not a critical point. This statement is false.

3 statement- The graph of f has a point of inflection at [tex]x = 1[/tex].

An inflection point occurs where [tex]f''(x) = 0[/tex]. [tex]f''(x)[/tex] is the slope of[tex]f'(x)[/tex] graphed above. Notice the slope is zero at[tex]x = 1[/tex] for the graph above. So[tex]f''(x) = 0[/tex] at[tex]x = 1[/tex]. There is an inflection point [tex]x = 1.[/tex]

So the correct option is statement I and III only.

For more details on graph of derivatives follow the link below:

https://brainly.com/question/2284808