Colt Rhodes

2022-04-10

To calculate:

$f\left(x\right)=-3{x}^{5}-2{x}^{4}-6{x}^{3}+x-7$ concave or convex at $x=5$ ?

ruseducatives1t03

Beginner2022-04-11Added 8 answers

Step 1

It's a pretty straightforward polynomial, so taking the derivative is pretty easy. You can just derive the terms one by one, proceeding left to right:

$\frac{d}{dx}-3{x}^{5}-2{x}^{4}-6{x}^{3}+x-7$

$=-15{x}^{4}-8{x}^{3}-18{x}^{2}+1$

And then, do it again:

$\frac{d}{dx}-15{x}^{4}-8{x}^{3}-18{x}^{2}+1$

$=-60{x}^{3}-24{x}^{2}-36x$

You can plug in$x=5$ and calculate if you want, but if you're pressed for time on an exam it's not really necessary. Each term in the second derivative is negative for $x=5$ , so the second derivative is therefore negative at that point, so it's convex (or concave downward) at $x=5$ .

It's a pretty straightforward polynomial, so taking the derivative is pretty easy. You can just derive the terms one by one, proceeding left to right:

And then, do it again:

You can plug in

Find the local maximum and minimum values and saddle points of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function

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