Question

Which one of the following options represents the given graph? 

• A. $f(x)=x^{2}\,2^{-|x|}$
• B. $f(x)=x\,2^{-|x|}$
• C. $f(x)=|x|\,2^{-x}$
• D. $f(x)=x\,2^{-x}$

Hint:

To match a graph, first infer symmetry (even/odd) and end behavior. Exponential decay like $2^{-|x|}$ forces both tails to $0$, while a factor $x^{2}$ ensures evenness and a zero at the origin.

Answer 1

The Correct Option is A

Step 1: Interpreting the graphical clues
The problem describes a curve that is symmetric about the y-axis, always nonnegative, dips to exactly 0 at x = 0, and has two equal peaks on either side before decaying towards zero as |x| increases. Such behavior strongly suggests that the function is even, satisfies f(0) = 0, remains ≥ 0 for all x, and decays for large |x|.

Step 2: Eliminate inconsistent options
- Option (B): \(x \, 2^{-|x|}\). This function is odd, since changing x to –x changes the sign. But the graph is symmetric and nonnegative, so this cannot match.

- Option (D): \(x \, 2^{-x}\). This is also odd and changes sign across the y-axis. Therefore, it is inconsistent with the graph as well.

- Option (C): \(|x| \, 2^{-x}\). Although the absolute value ensures nonnegativity for x ≥ 0, for x < 0 the factor \(2^{-x} = 2^{|x|}\) grows exponentially. Thus, instead of decaying as |x| increases, the function diverges to infinity for negative x. This contradicts the observed decaying tails on both sides.

Step 3: Test the remaining candidate (A)
Consider \(f(x) = x^{2} \, 2^{-|x|}\).
- Evenness: The function involves \(x^{2}\) and \(|x|\), both even in x, hence the function is symmetric about the y-axis.
- At the origin: \(f(0) = 0^{2} \cdot 2^{0} = 0\), matching the central dip to 0.
- Behavior at infinity: As \(|x| \to \infty\), the polynomial growth \(x^{2}\) is dominated by the exponential decay \(2^{-|x|}\), so the function tends to 0, matching the graph’s vanishing tails.
- Local shape: For small positive |x|, the quadratic term makes the function grow from 0, then the exponential decay takes over, creating a pair of symmetric peaks before returning towards 0. This precisely matches the description of the graph.

Step 4: Final confirmation
Since only option (A) satisfies all conditions (symmetry, nonnegativity, correct origin behavior, correct tail decay, and peak formation), it is the correct function.

Final Answer:
\[ \boxed{f(x) = x^{2} \, 2^{-|x|}} \]