For what values of x is \(\displaystyle{f{{\left({x}\right)}}}={3}{x}^{{{3}}}-{7}{x}^{{{2}}}-{5}{x}+{9}\)

kembdumatxf

kembdumatxf

Answered question

2022-03-27

For what values of x is f(x)=3x37x25x+9 concave or convex?

Answer & Explanation

ineditablesdmx0

ineditablesdmx0

Beginner2022-03-28Added 9 answers

Step 1
The convexity and concavity of the function f can be determined by looking at the sign of the second derivative:
If f>0, then f is convex.
If f<0, then f is concave.
To find the function's second derivative, use the power rule.
f(x)=3x37x25x+9
f(x)=9x214x5
f(x)=18x14
So, the convexity and concavity are determined by the sign of f(x)=18x14
The second derivative equals 0 when 18x14=0, which is at x=79
When x>79, f(x)>0, so f(x) is convex on (79, )
When x<79, f(x)<0, so f(x) is concave on (, 79)

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