Given f(x)=x^7 -x^5 -x^4 +2x +1 on interval [-1,1] show that there's at least one critical point on this interval.
The function "f(x)" is continuous and differentiable on "\\R" as polynomial.
"f'(x)=7x^6-5x^4-4x^3+2""f'(0)=2>0""f'(0.8)=7(0.8)^6-5(0.8)^4-4(0.8)^3+2"
"=-0.260992<0"
Then by the Intermediate Value Theorem, there exists a number "c\\in (0, 0.8)" such that "f'(c)=0."
Therefore there's at least one critical point on the interval"[-1,1]."
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