Question

Let $A(x) = \int_0^x f(t) dt$, with $f(x)$ shown in the graph below. At what $x$ value(s) does $A(x)$ have a local max: $x = $ [Use commas to separate multiple values.] At what $x$ value(s) does $A(x)$ have a local min: $x = $ [Use commas to separate multiple values.] Restrict your responses to values $0 < x < 6$.

          Let $A(x) = \int_0^x f(t) dt$, with $f(x)$ shown in the graph below.
At what $x$ value(s) does $A(x)$ have a local max: $x = $
[Use commas to separate multiple values.]
At what $x$ value(s) does $A(x)$ have a local min: $x = $
[Use commas to separate multiple values.]
Restrict your responses to values $0 < x < 6$.
        
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Let A(x) = ∫0^x f(t) dt, with f(x) shown in the graph below.
At what x value(s) does A(x) have a local max: x =
[Use commas to separate multiple values.]
At what x value(s) does A(x) have a local min: x =
[Use commas to separate multiple values.]
Restrict your responses to values 0 < x < 6.

Added by Frank C.

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Calculus: Early Transcendentals
Calculus: Early Transcendentals
James Stewart 8th Edition
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etA( f(t) dt, with f() shown in the graph below. At what value(s) does A() have a local max: [Use commas to separate multiple values.] At what value(s) does A() have a local min: [Use commas to separate multiple values.] Restrict vour responses to values 0<<6
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Transcript

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00:01 All right, so we're given the following graph, a of x equals 0 to x, the integral from 0 to x of f of x dt, where f of x is shown in the graph below.
00:18 We're going to determine where a of x has a local max.
00:22 What we're going to note here is that if we find the derivative of a, that gives us the function f of x.
00:29 So the graph of f of x below is actually the derivative of a.
00:34 So let's just think of it as the derivative of a, not as the function a, as the derivative.
00:39 So if this graph that we're given, which looks like this, is going to be the derivative of a, and we're given that the x -axis, it's something like this.
00:50 That's the x -axis, and then this is going to be the y right about here.
00:55 In order for us to have a local min or local max, i have to have a negative to a positive derivative to have a local min, and then a positive to a negative derivative to have a local max.
01:09 In order for this to be the case, i'm going to highlight where a prime, or the derivative of f, or in this case, a, is positive.
01:18 It's positive here, and it's positive here.
01:22 I'm now going to highlight where it's negative.
01:25 Negative there, negative there, negative there.
01:28 Wherever we go from a blue to a yellow, blue to a yellow indicates a negative to a positive...
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