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Question:

Consider the statements:
P: There exists some x ∈ R such that f(x) + 2x = 2(1 + x²)
Q: There exists some x ∈ R such that 2f(x) + 1 = 2x(1 + x)
Then
both P and Q are true
P is false and Q is true
P is true and Q is false
both P and Q are false

Pis true andQis false

bothPandQare true

bothPandQare false

Pis false andQis true

Solution:

f(x) = (1-x)²sin2x + x² for all x ∈ R
g(x) = ∫₁ˣ (2(t⁵)t + 1 - lnt)f(t)dt for all x ∈ (1, ∞)