Name: Local Maxima and Minima of a Function
Uploaded: 2019-07-19
Duration: 14 min 17 s
Description: Do you know what is minima or maxima of a function? This video also explains how to find local maxima and minima by differentiation in a detailed manner.

Question

Construct the graph of the function

y = f (x - 1) + f (x + 1)

, where

f (x) = \{\begin{cases} 1 - |x| & when |x| \leq 1 \\ 0 & when |x| > 1 \end{cases}

.

Accepted Answer

Given: $y = f (x - 1) + f (x + 1)$

Where, $f (x) = \{\begin{cases} 1 - |x|, |x| \leq 1 \\ 0, |x| > 1 \end{cases}$

Consider, $g (x) = |x| = \{\begin{cases} x, x \geq 0 \\ - x, x \leq 0 \end{cases}$

Domain is $x \in R$ .

Range is $[0, \infty)$ .

Consider, $h (x) = - |x| = \{\begin{cases} - x, x \geq 0 \\ x, x < 0 \end{cases}$

Domain is $x \in R$ .

Range is $(- \infty, 0]$ .

Consider, $y = p (x) = 1 - |x|$ .

$\Rightarrow p (x) = \{\begin{cases} 1 - x, x \geq 0 \\ 1 + x, x < 0 \end{cases}$

Graph of $h (x)$ will be shifted vertically upwards by $1$ unit (to draw the graph of $p (x)$ ).

$\Rightarrow f (x) = \{\begin{matrix} 1 - x & 0 \leq x < 1 \\ 1 + x & - 1 < x < 0 \\ 0 & | x | > 1 \end{matrix} \begin{array}{l} \end{array}$

So, the graph of $f (x)$ is

$y = f (x + 1) + f (x - 1)$

$f (x + 1) = \{\begin{cases} 1 - |x + 1|, |x + 1| \leq 1 \\ 0, |x + 1| > 1 \end{cases}$

For $|x + 1| \leq 1$

$\Rightarrow - 1 \leq x + 1 \leq 1$

$\Rightarrow - 2 \leq x \leq 0$

$\Rightarrow f (x + 1) = \{\begin{cases} \begin{matrix} 1 - (x + 1), - 1 \leq x \leq 0 = - x \\ 1 + (x + 1), - 2 \leq x \leq - 1 = 2 + x \end{matrix} \\ 0, x \in (- \infty, - 2) \cup (0, \infty) \end{cases}$

$f (x - 1) = \{\begin{cases} 1 - |x - 1|, |x - 1| < 1 \\ 0, |x - 1| > 1 \end{cases}$

$\Rightarrow f (x - 1) = \{\begin{cases} \begin{matrix} 1 - (x - 1), 1 \leq x \leq 2 = 2 - x \\ 1 + (x - 1), 0 \leq x \leq 1 = x \end{matrix} \\ 0, x \in (- \infty, 0) \cup (2, \infty) \end{cases}$

So, $y = f (x - 1) + f (x + 1)$ .

$\Rightarrow y = \{\begin{cases} \begin{matrix} 0, x \in (- \infty, - 2) \cup (2, \infty) \\ 2 + x, x \in [- 2, - 1) \\ - x, x \in [- 1, 0] \\ x, x \in (0, 1] \end{matrix} \\ 2 - x, x \in [1, 2] \end{cases}$

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