The function
𝑓
(
𝑥
)
=
1
𝑒
2
𝑥
−
𝑥
2
is concave for

Question

The function 
𝑓
(
𝑥
)
=
1
𝑒
2
𝑥
−
𝑥
2
    is concave for

BlackTom AI · Accepted Answer

We are given a function f(x) with a request about where it is concave, and the available option is a single statement: x > ln(2).

First, interpret the likely form of the function from the prompt: f(x) = e^{2x - x^2} (this is a common way to present a function in such questions, written compactly as f(x) = e^{g(x)} with g(x) = 2x - x^2).

To determine concavity, we examine the second derivative. If f(x) = e^{g(x)}, then f'(x) = f(x) g'(x) and f''(x) = f(x) [(g'(x))^2 + g''(x)]. Since e^{g(x)} > 0 for all x, the sign of f''(x) is governed by (g'(x))^2 + g''(x).

Compute g'(x) and g''(x):
- g'(x) = d/dx (2x - x^2) = 2 - 2x
- g''(x) = d/dx (2 - 2x) = -2

Plug into the second derivative factor:
(g'(x))^2 + g''(x) = (2 - 2x)^2 + (-2) = 4(1 - x)^2 - 2.

Set this expression ≤ 0 to find concavity (f''(x) ≤ 0):
4(1 - x)^2 - 2 ≤ 0
=> 4(1 - x)^2 ≤ 2
=> (1 - x)^2 ≤ 1/2
=> |x - 1| ≤ 1/√2.

This yields the interval where f is concave: x ∈ [1 - 1/√2, 1 + 1/√2], which numerically is about [0.293, 1.707].

Now compare with the given option: 'x > ln(2)'. Since ln(2) ≈ 0.693, the statement x > ln(2) asserts concavity for all x greater than about 0.693. However, the actual concavity interval is centered around x = 1 and extends only from approximately 0.293 to 1.707; it does not simply correspond to 'x > ln(2)'. In particular, concavity holds for some x less than ln(2) (e.g., x ≈ 0.5 is inside the concave interval), and concavity ceases again for x > about 1.707. Therefore, the statement 'x > ln(2)' is not the correct description of the concavity region for f.

If the function were different (for example, f(x) = e^{2x} - x^2 or f(x) = e^{-2x} - x^2, etc.), the concavity analysis would change accordingly because the expression for f'' would be different. But given the typical interpretation f(x) = e^{2x - x^2}, the conclusion above applies, and the option 'x > ln(2)' does not correctly specify the concavity interval.

ECON5005 (ND) Quiz 3 - 2025S1

The function 𝑓 ( 𝑥 ) = 1 𝑒 2 𝑥 − 𝑥 2 is concave for

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