Let g(x)=∫0^e^x (f^'(t))/(1+t^2) d t . Which of the...

Let $g(x)=\int_{0}^{e^{x}} \frac{f^{\prime}(t)}{1+t^{2}} d t .$ Which of the following is true? (a) $g^{\prime}(x)$ is positive on $(-\infty, 0)$ and negative on $(0, \infty)$ (b) $g^{\prime}(x)$ is negative on $(-\infty, 0)$ and positive on $(0, \infty)$ (c) $g^{\prime}(x)$ change sign on both $(-\infty, 0)$ and $(0, \infty)$ (d) $g^{\prime}(x)$ does not change sign on $(-\infty, \infty)$

Let $g(x)=\int_{0}^{e^{x}} \frac{f^{\prime}(t)}{1+t^{2}} d t .$ Which of the following is true?
(a) $g^{\prime}(x)$ is positive on $(-\infty, 0)$ and negative on $(0, \infty)$
(b) $g^{\prime}(x)$ is negative on $(-\infty, 0)$ and positive on $(0, \infty)$
(c) $g^{\prime}(x)$ change sign on both $(-\infty, 0)$ and $(0, \infty)$
(d) $g^{\prime}(x)$ does not change sign on $(-\infty, \infty)$

Key Concepts

Properties of the Exponential Function

The exponential function has the characteristic of being always positive for any real input. This property is important in any analysis involving exponentials, particularly when they are composed with other functions, because they do not alter the sign of the overall expression.

Sign Analysis of Functions

Sign analysis is the process of determining where a function, or its derivative, is positive or negative by analyzing the individual factors within the function. This technique is crucial when the derivative is expressed as a product of functions, as it helps in identifying intervals of increase or decrease.

Leibniz Integral Rule

The Leibniz integral rule is a generalization used for differentiating an integral whose limits and integrand both may depend on the variable of differentiation. It accounts for changes in both the limits of integration and the integrand.

Chain Rule

The chain rule is used to differentiate composite functions. When the upper limit of an integral is a function of the variable, the chain rule implies that the derivative of the integral involves the derivative of the upper limit multiplied by the evaluation of the integrand at that limit.

Fundamental Theorem of Calculus

This theorem establishes the connection between integration and differentiation. It states that if a function is defined as an integral with a variable limit, the derivative of this function is obtained by evaluating the integrand at the limit and then multiplying by the derivative of that limit.

Please give Ace some feedback