Is \(\displaystyle{f{{\left({x}\right)}}}=-{2}{x}^{{{5}}}-{2}{x}^{{{4}}}+{5}{x}-{45}\) concave or convex at

Terrence Riddle

Terrence Riddle

Answered question

2022-04-14

Is f(x)=2x52x4+5x45 concave or convex at x=2?

Answer & Explanation

libertydragonrbha

libertydragonrbha

Beginner2022-04-15Added 15 answers

Step 1
A necessary and sufficient condition for f to be convex on an interval is that f(x)>0 for all x in the interval.
In this case,
f(x)=40x324x2
So,
f(2)=40(8)24(4)<0
f(x) is continuous near -2, so f(x)<0 for x near -2 and f is convex near -2.
(If you have been given a definition of f is "convex at a number, a", then I would guess you'll say f is convex at -2.)

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