GATE CS 2015 Set 2 Q.3 (GA Q.3): Quantitative Aptitude

Consider a function $f(x) = 1 - |x|$ on $-1 \le x \le 1$ . The value of $x$ at which the function attains a maximum, and the maximum value of the function are:

$0, -1$

$-1, 0$

$0, 1$

$-1, 2$

Official answer and solution

Official answer: C
From: Official GATE 2015 answer key

Solution

Analyze the absolute value function $f(x) = 1 - |x|$ over the closed interval $[-1, 1]$ .

For any real number $x$ , the absolute value satisfies $|x| \ge 0$ , with equality $|x| = 0$ holding if and only if $x = 0$ .

Since $f(x) = 1 - |x|$ , subtracting a non-negative quantity $|x|$ from 1 reaches its maximum value when the subtracted quantity is minimized.

Thus, the minimum of $|x|$ is 0, achieved at $x = 0$ .

Evaluating at $x = 0$ gives:

f(0) = 1 - |0| = 1.

For any other point in $[-1, 1]$ , $|x| > 0$ , so $f(x) < 1$ . Thus, the maximum is attained at $x = 0$ , and the maximum value is 1.

Final answer

Common trap

Confusing the point $x$ at which the maximum occurs ( $x = 0$ ) with the maximum value ( $1$ ).

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

Source

Paper: GATE 2015 Computer Science and Information Technology (CS) Session 2
Question: 3 of 65, printed as GA Q.3
Section: General Aptitude

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