Functions Question - None

Question

For each real

x

, let f(x) = $\max\{x^{2}, x^{3}, x^{4}\}$, then f(x) is -

Accepted Answer

Step 1: Analyze for x ≤ -1

For x ≤ -1, all values are negative or x = -1. Let x = -2:

x² = 4
x³ = -8
x⁴ = 16

Here x⁴ > x² > x³. For all x ≤ -1, we have x⁴ ≥ x² because |x| ≥ 1 implies x⁴ ≥ x². Also x⁴ > x³ since x⁴ is positive and x³ is negative for x < 0. Therefore f(x) = x⁴ for x ≤ -1.

Step 2: Analyze for -1 < x ≤ 0

For -1 < x < 0: x² is positive, x³ is negative, x⁴ is positive. Since 0 < |x| < 1, we have |x|⁴ < |x|². This means x⁴ < x². So f(x) = x² for -1 < x ≤ 0.

Step 3: Analyze for 0 < x < 1

For 0 < x < 1: all three are positive. Since x < 1, we have x² > x³ and x² > x⁴. Therefore f(x) = x² for 0 < x < 1.

Step 4: Analyze for x ≥ 1

For x ≥ 1: x⁴ ≥ x² ≥ x³ (since x⁴ = x² · x² ≥ x² when x ≥ 1). Therefore f(x) = x⁴ for x ≥ 1.

Step 5: Verify boundary cases

At x = -1: x² = 1, x³ = -1, x⁴ = 1, so max = 1 = x⁴ ✓

At x = 0: all equal 0 ✓

At x = 1: x² = 1, x³ = 1, x⁴ = 1, all equal ✓

Step 6: Check option (C)

f(1/2) = max{(1/2)², (1/2)³, (1/2)⁴} = max{1/4, 1/8, 1/16} = 1/4 = (1/2)², not 1/2.

Step 7: Conclusion

Option (A) states f(x) = x⁴ for x ≤ -1, which is correct based on our analysis.

∴ Answer: A

Question:

Step-by-Step Solution

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