Answered: Find the asymptotes, intervals of…

College Algebra

1st Edition

ISBN:9781938168383

Author:Jay Abramson

Publisher:Jay Abramson

Chapter3: Functions

Section: Chapter Questions

Problem 30RE: For the following exercises, use the graphs to determine the intervals on which the functions are...

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Question

Find the asymptotes, intervals of increase, decrease, concave upward,
concave downward, local max, local min, of function f(x) = x +√(1 − x) .

Expert Solution

Step 1

$f (x) = x + \sqrt{1 - x} f' (x) = 1 - \frac{1}{2 \sqrt{1 - x}} f o r i n c r e a \sin g, f' (x) \geq 0 \Rightarrow 1 - \frac{1}{2 \sqrt{1 - x}} \geq 0 \Rightarrow \frac{1}{2 \sqrt{1 - x}} \leq 1 \Rightarrow \sqrt{1 - x} \geq \frac{1}{2} \Rightarrow 1 - x \geq \frac{1}{4} \Rightarrow x \leq \frac{3}{4} i n t e r v a l s f o r i n c r e a \sin g o f f u n c t i o n i s (- \infty, \frac{3}{4}] f o r d e c r e a \sin g, f' (x) \leq 0 \Rightarrow 1 - \frac{1}{2 \sqrt{1 - x}} \leq 0 \Rightarrow \frac{1}{2 \sqrt{1 - x}} \geq 1 \Rightarrow \sqrt{1 - x} \leq \frac{1}{2} \Rightarrow 1 - x \leq \frac{1}{4} \Rightarrow x \geq \frac{3}{4} i n t e r v a l f o r d e c r e a \sin g [\frac{3}{4}, \infty) f o r c r i t i c a l p o i n t o f f' = 0 1 - \frac{1}{2 \sqrt{1 - x}} = 0 \Rightarrow 2 \sqrt{1 - x} - 1 = 0 \Rightarrow 2 \sqrt{1 - x} = 1 \Rightarrow 4 (1 - x) = 1 \Rightarrow 1 - x = \frac{1}{4} \Rightarrow x = \frac{3}{4} f " (x) = - \frac{1}{4 \sqrt[3]{1 - x}} f " (\frac{3}{4}) = - 2 < 0 t h e r e f o r e, f h a s l o c a l m a x i m a a t x = \frac{3}{4} a n d f i s c o n c a v e d o w n w a r d$

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