Question

Which of the extrema below does the the function f(x) = x³ − 2x³ + 1 have? 1. absolute maximum II. absolute minimum III. local maximum IV. local minimum ACTS A. I and Ill only B. I, II, III, and IV OC. I and II only OD. III and IV only Portions of this soft

Answer 1

`f\left(x\right)\:=\:x^5-2x^3+1`
`f^(\prime)(x)=5x^4-6x^2`
`\mathrm{Suppose\:that\:}x=c\mathrm{\:is\:a\:critical\:point\:of\:}f\left(x\right)\mathrm{\:then,\:}`
`\mathrm{If\:}f\:'\left(x\right)>0\mathrm{\:to\:the\:left\:of\:}x=c\mathrm{\:and\:}f\:'\left(x\right)<0\mathrm{\:to\:the\:right\:of\:}x=c\mathrm{\:then\:}x=c\mathrm{\:is\:a\:local\:maximum.}`
`\mathrm{If\:}f\:'\left(x\right)<0\mathrm{\:to\:the\:left\:of\:}x=c\mathrm{\:and\:}f\:'\left(x\right)>\:0\mathrm{\:to\:the\:right\:of\:}x=c\mathrm{\:then\:}x=c\mathrm{\:is\:a\:local\:minimum.}`

`f^(\prime)(x)=5x^4-6x^2=0\text{ }\rightarrow x\text{ }^2(5x^2\text{ - 6})=0`

then at x = 0 and x = (6/5)^(1/2) the function has critical points. Minding the sign of f', one obtains that there are only local maximum and local minimum, so the answer is D.