Difference between revisions of "1982 AHSME Problems/Problem 19"
MRENTHUSIASM (talk | contribs) (Created page with "== Problem == Let <math>f(x)=|x-2|+|x-4|-|2x-6|</math> for <math>2 \leq x\leq 8</math>. The sum of the largest and smallest values of <math>f(x)</math> is <math>\textbf {(A)...") |
MRENTHUSIASM (talk | contribs) (→Solution) |
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\end{cases}.</cmath> | \end{cases}.</cmath> | ||
The graph of <math>y=f(x)</math> is shown below. | The graph of <math>y=f(x)</math> is shown below. | ||
+ | <asy> | ||
+ | /* Made by MRENTHUSIASM */ | ||
+ | size(220); | ||
+ | import TrigMacros; | ||
− | + | rr_cartesian_axes(-2,10,-2,4,useticks=true); | |
+ | pair A[]; | ||
+ | A[0] = (0,0); | ||
+ | A[1] = (2,0); | ||
+ | A[2] = (3,2); | ||
+ | A[3] = (4,0); | ||
+ | A[4] = (8,0); | ||
+ | |||
+ | draw(A[0]--A[1]--A[2]--A[3]--A[4],red+linewidth(1.5)); | ||
+ | |||
+ | for(int i = 0; i <= 4; ++i) { | ||
+ | dot(A[i],red+linewidth(4.5)); | ||
+ | } | ||
+ | |||
+ | label("$(0,0)$",A[0],(0,-1.5),UnFill); | ||
+ | label("$(2,0)$",A[1],(0,-1.5),UnFill); | ||
+ | label("$(3,2)$",A[2],(0,1.5),UnFill); | ||
+ | label("$(4,0)$",A[3],(0,-1.5),UnFill); | ||
+ | label("$(8,0)$",A[4],(0,-1.5),UnFill); | ||
+ | </asy> | ||
The largest value of <math>f(x)</math> is <math>2,</math> and the smallest value of <math>f(x)</math> is <math>0.</math> So, their sum is <math>\boxed{\textbf {(B)}\ 2}.</math> | The largest value of <math>f(x)</math> is <math>2,</math> and the smallest value of <math>f(x)</math> is <math>0.</math> So, their sum is <math>\boxed{\textbf {(B)}\ 2}.</math> | ||
Revision as of 21:17, 12 September 2021
Problem
Let for
. The sum of the largest and smallest values of
is
Solution
Note that at one of the three absolute values is equal to
Without using absolute values, we rewrite as a piecewise function:
which simplify to
The graph of
is shown below.
The largest value of
is
and the smallest value of
is
So, their sum is
~MRENTHUSIASM
See Also
1982 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 18 |
Followed by Problem 20 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 | ||
All AHSME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.