# 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
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.

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.

### Relevant Questions

Find the absolute max and min values at the indicated interval
$$f(x)=2x^{3}-x^{2}-4x+10, [-1,0]$$
Find the absolute maximum and absolute minimum values of f on the given interval and state where those values occur: $$f(x)=x^{3}-3x^{2}-9x+25, [-5,10]$$
Find the absolute maximum and minimum values of f on the given interval.
$$f(x)=4x^{3}-6x^{2}-24x+9. [-2,3]$$
Find the absolute maximum and absolute minimum values of f over the interval. $$f(x)=(\frac{4}{x})+\ln(x^{2}), 1\leq x\leq 4$$
Consider the function $$f(x)=2x^{3}-6x^{2}-18x+9$$ on the interval [-2,4].
What is the absolute minimum of f(x) on [-2,4]?
What is the absolute maximum of f(x) on [-2,4]?
Find the absolute maximum and absolute minimum values of f on the given interval.
$$f(x)=x+\frac{4}{x},[0.2,8]$$
The graph of y = f(x) contains the point (0,2), $$\displaystyle{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}}={\frac{{-{x}}}{{{y}{e}^{{{x}^{{2}}}}}}}$$, and f(x) is greater than 0 for all x, then f(x)=
A) $$\displaystyle{3}+{e}^{{-{x}^{{2}}}}$$
B) $$\displaystyle\sqrt{{{3}}}+{e}^{{-{x}}}$$
C) $$\displaystyle{1}+{e}^{{-{x}}}$$
D) $$\displaystyle\sqrt{{{3}+{e}^{{-{x}^{{2}}}}}}$$
E) $$\displaystyle\sqrt{{{3}+{e}^{{{x}^{{2}}}}}}$$
$$f(x)=5+54x-2x^{3}, [0,4]$$
The function $$f(x)=4-7x^{4}$$ has an absolute maximum value of ? and this occurs at x equals ?