Question

Define f: ℝ → ℝ and g: ℝ → ℝ as follows
$f(x)=\sum\limits^{\infin}_{m=0}\frac{(-1)^mx^{2m}}{2^{2m}(m!)^2}$ and $g(x)=\frac{x}{2}\sum\limits^{\infin}_{m=0}\frac{(-1)^mx^{2m}}{2^{2m}(m+1)!m!}$ for x ∈ $\R$.
Let x1, x2, x3, x4 ∈ ℝ be such that 0 < x1 < x2 , 0 < x3 < x4,
f(x1) = f(x2) = 0, f(x) ≠ 0 when x1 < x < x2,
g(x3) = g(x4) = 0 and g(x) ≠ 0 when x3 < x < x4.
Then, which of the following statements is/are TRUE ?

• A. The function f does not vanish anywhere in the interval (x3, x4)
• B. The function f vanishes exactly once in the interval (x3, x4)
• C. The function g does not vanish anywhere in the interval (x1, x2)
• D. The function g vanishes exactly once in the interval (x1, x2)

Answer 1

Answer: (B), (D)

The function f vanishes exactly once in the interval (x3, x4)

Answer 2

The correct option is (B) : The function f vanishes exactly once in the interval (x3, x4) and (D) : The function g vanishes exactly once in the interval (x1, x2).