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.

Question
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.
0

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