Tyra

2021-08-17

In each item, do the following analytically: (a) find the relative extrema of f. (b) determine the values of x at which the relative extrema occur. (c) determine the intervals on which f is increasing. (d) determine the intervals on which f is decreasing. $f\left(x\right)=2-{\left(x-1\right)}^{\frac{1}{3}}$

unett

(a) Find the domain of the funtion Find the derivative of the function and the domain of the derivetive ${f}^{\prime }\left(x\right)=\frac{1}{3\sqrt[3]{\left(x-1{\right)}^{2}}}$
Substitute f'(x)=0 ti find the critical points and solve for x $0=\frac{1}{3\sqrt[3]{\left(x-1{\right)}^{2}}}$
$x\in \varnothing$ Since there is no value of x that makes the derivative of the funtion 0, the function has no relative exterma (b) To find x-intercept, substitute f(x)=0 $0=-2-{\left(x-1\right)}^{\frac{1}{3}}$ Move expression to th left-hand side and change its sign ${\left(x-1\right)}^{\frac{1}{3}}=2$ Tranform and simplify the quation $\sqrt[3]{\left(x-1\right)}=2$ Raise both sides of the equation to th power of 3 $x-1=8$ Move constant to th right side and change its sign, than calculate $x=8+1$
$x=9$ (c) and (d) Let's plot the polynomial to determine the intervals of increasing and decreasing. Decreases by: $\left(-\mathrm{\infty },1\right),\left(1,\mathrm{\infty }\right)$

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