Let f(x)=e^v+1 / n-e^x for real number p>0. The value of...

Let $f(x)=e^{v+1 / n}-e^{x}$ for real number $p>0$.
The value of $x=s$, for which $f(x)$ is minimum, is
(A) $-\frac{\ln (\mathrm{p}+1)}{\mathrm{p}}$
(B) $-\ln (\mathrm{p}+1)$
(C) $-\ln \mathrm{p}$
(D) $\ln \left(\frac{\mathrm{p}+\mathrm{l}}{\mathrm{p}}\right)$

Key Concepts

Logarithmic Transformation

Logarithmic transformation is a useful algebraic technique to solve equations where the unknown variable appears in the exponent, such as those arising from differentiation of exponential functions. By taking the natural logarithm of both sides of an equation, one can simplify and linearize the problem, making it easier to isolate and solve for the variable.

Exponential Functions

Exponential functions are functions of the form exp(x) or e^(x). They have the unique property that their derivative is proportional to the function itself. This property is key in optimization problems involving exponentials because it often leads to equations that can be solved by logarithmic transformations.

Critical Points

Critical points of a function are those values of the independent variable where the derivative is zero or does not exist. These points are potential candidates for local minima or maxima, and further analysis, such as the second derivative test, is used to classify their exact nature.

Differentiation

Differentiation is the process of computing the derivative of a function, which measures its instantaneous rate of change. In a typical optimization problem, the derivative is used to locate critical points by solving for when the derivative equals zero, meaning the slope of the function’s tangent is horizontal.

Optimization

Optimization involves finding the extreme values (minimum or maximum) of a function by identifying points where the function’s behavior changes. In calculus, this generally involves looking for critical points where the first derivative is zero or undefined, and then using tests (such as the second derivative test) to determine if these points yield a minimum or maximum.

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