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# What values(s) of the constant b make f(x)=x^{3}−bx for 0leq xleq 2 have: a.) an absolute min at x=1? Explain. b.) an absolute max at x=2? Explain. # What values(s) of the constant b make f(x)=x^{3}−bx for 0leq xleq 2 have: a.) an absolute min at x=1? Explain. b.) an absolute max at x=2? Explain.

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Functions asked 2021-02-13
What values(s) of the constant b make $$f(x)=x^{3}−bx$$ for $$0\leq x\leq 2$$ have:
a.) an absolute min at x=1? Explain.
b.) an absolute max at x=2? Explain.

## Answers (1) 2021-02-14
(a). Need to find b so that $$f(x)=x3−bxf(x)=(x^{3})−bx$$ has minimum value at x=1.
Here, $$f'(x) = (3x^{2})-b$$. Now f'(1) = 0 gives us $$3-b=0 \Rightarrow b=3$$
Here we have
$$f''(x)=6x\Rightarrow f''(1)=6,$$
therefore, ff has an absolute min at x=1.
(b). We have $$f'=3(x^{2})-b$$, so f is increasing whenever b is non-negative, In that case, f is increasing in $$0\leq x\leq2$$ and attains its absolute maximum at x=2.

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