Is \(\displaystyle{f{{\left({x}\right)}}}={\left({x}-{3}\right)}{\left({x}+{2}\right)}{\left({x}-{4}\right)}^{{{2}}}\) concave or convex at

Kale Bright

Kale Bright

Answered question

2022-04-14

Is f(x)=(x3)(x+2)(x4)2 concave or convex at x=1?

Answer & Explanation

awalkbyfaithbzu6

awalkbyfaithbzu6

Beginner2022-04-15Added 21 answers

Step 1
We need
(uvw)=uvw+vuw+wuv
We must calculate the first and second derivatives
f(x)=(x3)(x+2)(x4)2
f(x)=(x+2)(x4)2+(x3)(x4)2+2(x3)(x+2)(x4)
=(x4){(x+2)(x4)+(x3)(x4)+2(x3)(x+2)}
=(x4){x22x8+x27x+12+2x22x12}
=(x4)(4x211x8)
f(x)=(4x211x8)+(x4)(8x11)
=4x211x8+9x243x+44
=12x254x+36
Therefore,
f(1)=12+54+36=102
As, f(1)>0, we conclude that the function is convex

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