Maximum Value of the Function f(x) = 1 − e^x on the Real Line

Introduction

To find the maximum value of the function f(x) = 1 − e^x for all real numbers,
it is essential to understand how the exponential function e^x behaves as x moves towards positive
and negative infinity. Using this behavior, we can determine the extreme values of f(x) and then check which
of the given multiple-choice options is correct.

Problem Statement

Find the maximum value of the function:

f(x) = 1 − e^x, for −∞ < x < ∞

Options:

• A. ∞
• B. 2
• C. 1
• D. 0

Step-by-Step Solution

1. Consider the behavior of the exponential function e^x as x → +∞.
   In this case, e^x → ∞, so:

   f(x) = 1 − e^x → 1 − ∞ = −∞.

   This means that as x increases without bound, the function f(x) decreases without bound and
   does not have any upper bound coming from the right side.

2. Now consider x → −∞. In this case, e^x → 0⁺, so:

   f(x) = 1 − e^x → 1 − 0 = 1.

   Therefore, the function approaches the value 1 from below as x becomes very large negative,
   but it never exceeds 1.

3. For any real x, e^x is always positive. Hence:

   e^x > 0 ⇒ 1 − e^x < 1.

   So f(x) is always strictly less than 1, though it can be made arbitrarily close to 1 by taking
   x sufficiently negative. This means 1 is the supremum (least upper bound) of the function.

In many exam settings, when asked for the “maximum value” over (−∞, ∞) for such a function,
the intended answer is this supremum, which is 1.

Analysis of Each Option

Option A: ∞

This option would mean that the function grows without bound above. However,
f(x) = 1 − e^x never exceeds 1 and instead decreases to −∞ as x → +∞.
Therefore, ∞ cannot be the maximum value.

Option B: 2

For f(x) to be 2, we would need:

1 − e^x = 2 ⇒ e^x = −1.

Since e^x is always positive for real x, there is no real solution to e^x = −1.
Thus the function never reaches or approaches the value 2, making this option incorrect.

Option C: 1

As x → −∞, e^x → 0⁺ and therefore f(x) → 1.
The function values can get arbitrarily close to 1 but never exceed it.
Hence 1 is the supremum (greatest value approached) of f(x) on the entire real line.

Among the given discrete options, 1 is the correct choice for the maximum value.

Option D: 0

The value 0 occurs when:

1 − e^x = 0 ⇒ e^x = 1 ⇒ x = 0.

So, f(0) = 0. However, there exist values of x (large negative values) for which f(x) is closer to 1,
which is greater than 0. Hence, 0 is not the maximum value of the function.

Final Answer

The maximum value (supremum) of the function f(x) = 1 − e^x for −∞ < x < ∞ is:

1

Therefore, the correct option is C. 1.