Use of 2nd order derivatives

[Skelly4444]

In my maths book, all the worked examples go through the process of differentiating the function twice.

From this point, the result is then set to greater than zero or less than zero, depending on whether we're trying to establish convex or concave etc.

The worked example I have attached has confused me somewhat as I can't seem to see what they've done here?

Surely, if you set the 2nd order derivative to greater than zero here, you obtain a result of -0.5 on the right hand side of the inequality?

Any advice would be appreciated.

 

[Dr.Peterson]

In my maths book, all the worked examples go through the process of differentiating the function twice.

From this point, the result is then set to greater than zero or less than zero, depending on whether we're trying to establish convex or concave etc.

The worked example I have attached has confused me somewhat as I can't seem to see what they've done here?

Surely, if you set the 2nd order derivative to greater than zero here, you obtain a result of -0.5 on the right hand side of the inequality?

Any advice would be appreciated.

The problem is:

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The work is:

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They aren't solving an inequality; they are showing that it is always true. Since the exponential is always positive, so is 4 times that, and 2 more than that. Therefore it is always true that [imath]4e^{2x}+2>0[/imath].

If you did try to solve it (to find when it is true), you would find that it is true whenever [imath]e^{2x}>-0.5[/imath], which is true for all x, so you get the same result. But it isn't necessary to do that.