How do you determine whether the function \(\displaystyle{F}{\left({x}\right)}={\frac{{{1}}}{{{12}}}}{X}^{{{4}}}+{\frac{{{1}}}{{{6}}}}{X}^{{{3}}}-{3}{X}^{{{2}}}-{2}{X}+{1}\) is

Lisa Cooper

Lisa Cooper

Answered question

2022-04-17

How do you determine whether the function
F(x)=112X4+16X33X22X+1
is concave up or concave down and its intervals?

Answer & Explanation

szalbierzfytg

szalbierzfytg

Beginner2022-04-18Added 13 answers

Step 1
For,
F(x)=112x4+16x33x22x+1
We have
F(x)=13x3+12x26x2
and
F(x)=x3+x6=(x+3)(x2)
The only chance F(x) has to (perhaps) change sign is at F(x)=0
Which happens at x=3 and at x=2
On (, 3), both factor of F are negative, so F is positive and the graph of f is concave up.
On (3, 2), we get F(x) is negative, so the graph is concave down.
On (2, , F(x) is positive, so the graph is concave up.
The graph of F is concave up on (, 3) and on (2, ) and it is concave down on (3, 2)

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