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

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Let f(x)=e^v+1 / n-e^x for real number p>0. Let g(t)=∫1^1+1...

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

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

Key Concepts

Differentiation Under the Integral Sign

Differentiation under the integral sign is a technique that allows one to differentiate an integral whose limits or integrand depend on a parameter. This method is particularly effective in problems where the optimum value of a parameter is sought by treating the integral as a function of that parameter and then differentiating it to find critical points.

Exponential Functions

Exponential functions are mathematical expressions in which a constant base is raised to a variable exponent. They possess unique properties such as a constant relative rate of change, which make them fundamental in solving differential equations, modeling growth and decay, and in various transformations within calculus. Their inverse functions, the logarithms, are often used to simplify and solve equations involving exponents.

Definite Integration

Definite integration involves computing the integral of a function over a specific interval, yielding a finite value that represents the net accumulation of the quantity described by the integrand. In problems involving exponential functions, integration techniques often lead to expressions that can be manipulated using properties of exponentials and logarithms, which are key in evaluating and simplifying the results.

Optimization in Calculus

Optimization in calculus is the process of finding the minimum or maximum values of a function, typically by determining where its derivative equals zero and then verifying the nature of these critical points. This concept is crucial for identifying extremal values of functions that depend on certain parameters, such as minimizing or maximizing the outcome of an integral with respect to a variable.

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