Question

If \(y = max \{ {\sqrt{x}, x^2-4, x^3+2}\}\), then the number of solution(s) of y=1 is/are____?

Answer 1

Given:
y = max { √x, x² − 4, x³ + 2 }

We need to find number of solutions of y = 1

Step 1: Condition for max = 1
At least one function = 1 and others ≤ 1

Step 2: Solve individually

(a) √x = 1 ⇒ x = 1
Check others at x = 1:
x² − 4 = 1 − 4 = −3 ≤ 1 ✔
x³ + 2 = 1 + 2 = 3 > 1 ✖
→ max = 3 ≠ 1 ❌

(b) x² − 4 = 1 ⇒ x² = 5 ⇒ x = ±√5
Domain: √x defined ⇒ x ≥ 0 ⇒ x = √5
Check at x = √5:
√x = √(√5) < 1 ✔
x³ + 2 > 1 ✖
→ max > 1 ❌

(c) x³ + 2 = 1 ⇒ x³ = −1 ⇒ x = −1
But √x not defined ❌

Step 3: Try inequality condition (all ≤ 1 and one = 1)

From x³ + 2 ≤ 1 ⇒ x ≤ −1
But √x requires x ≥ 0
→ No common region ❌

Step 4: Check boundary x = 0
√x = 0, x² − 4 = −4, x³ + 2 = 2
→ max = 2 ≠ 1 ❌

Final Answer: 0 solution