Answer to Question #158008 in Real Analysis for Cypress

Question

Give an upper bound and a lower bound for the expression 1/(a^(4)+3a^(2)+1) if a∈R

Answer 1

Solution

Let f(x) = 1/(x⁴+3x²+1)

Domain of f(x) is (-∞; ∞)

f(x) is continuous, f(x) >0 for x∈ (-∞; ∞)

f(x) is symmetric about y-axis

lim_|x|->∞ f(x) = 0 => axis Ox is asymptote of f(x)

df/dx = -(4x³+6x)/(x⁴+3x²+1) => df/dx = 0 only when x=0

d²f/dx² = -((12x²+6) (x⁴+3x²+1)- (4x³+6x)²)/(x⁴+3x²+1)² ; for x = 0  d²f/dx² = -6 <0 => x = 0 is the point of maximum

f(0) = 1

So range of f(x) is 0 < y ≤ 1

Therefore lower bound is 0, upper bound is 1

Answer

lower bound is 0, upper bound is 1