00:02
For a certain function, g of x, the derivative is given in the picture below.
00:08
So the graph is not of g, but of g.
00:10
What is the approximate slope of the tangent line to g of x at the point 2 comma g of 2? so the slope of the tangent line is going to be equal to g prime at that x value.
00:25
So the slope of the tangent is g prime of 2.
00:37
Now this is a graph of g prime.
00:39
At what's the value at 2, so at 2 or about here, so that looks to be about 1 .5.
00:46
Okay, so from the graph, we estimate g prime of 2 is about 1 1 .5.
01:10
So the slope is about 1 .5.
01:21
Okay, next we want to know how many real number solutions can there be to the equation g of x is 0.
01:26
Justify your conclusion fully and carefully by explaining what you know about how the graph of g must behave based on the graph of g.
01:33
Okay, so notice that g prime of x is always positive in the graph.
01:52
So what does that mean? that means that g is always increasing, thus there can be at most one solution to g of x equals zero.
02:28
Okay, now i'm basing this conclusion only on the interval in which the graph is shown.
02:33
Don't really know what's happening at the ends here.
02:35
I don't know if it's going to continue down forever or anything like that.
02:39
So i'm really only talking about the interval negative 4 to 4.
02:43
So on the shown interval negative 4 to 4.
02:57
Because if i'm increasing and i cross, you know, zero, then i cannot get back down to zero.
03:05
I would have to start decreasing at that point.
03:08
And this is a strictly increasing function.
03:10
Based on the graph of the derivative here.
03:15
Okay.
03:16
Next, on the interval from negative 3 to 3, how many times is the concavity of g change? okay, so g double prime positive means this concave up.
03:40
This occurs when? well, what does that mean? that means g prime prime is positive.
03:52
So the derivative of the derivative is positive...