Explained: "To calculate:f(x)=−3x5−2x4−6x3+x−7 concave or convex at x=5?"

High School
Calculus and Analysis
Calculus 1
Analyzing functions

Colt Rhodes

Answered question

2022-04-10

To calculate:

f (x) = - 3 x^{5} - 2 x^{4} - 6 x^{3} + x - 7

concave or convex at

x = 5

Answer & Explanation

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

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

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

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