If f(x) is a differentiable function and ϕ(x) is twice differentiable function and αand βare roo

step 1 So let's solve this question. So in this question we have let let G of X is equal to Y -dash -X,

If f(x) is a differentiable function and ϕ(x) is twice...

If $\mathrm{f}(\mathrm{x})$ is a differentiable function and $\phi(\mathrm{x})$ is twice differentiable function and $\alpha$ and $\beta$ are roots of the equation $f(x)=0$ and $\phi^{\prime}(x)=0$ respectively, then which of the following statement is true ? $(\alpha<\beta)$ (A) there exists exactly one root of the equation $\phi^{\prime}(x)$. $\mathrm{f}(\mathrm{x})+\phi^{\prime \prime}(\mathrm{x}) \cdot \mathrm{f}(\mathrm{x})=0$ on $(\alpha, \beta)$ (B) there exists atleast one root of the equation $\phi^{\prime}(x)$. $f(x)+\phi^{\prime \prime}(x) \cdot f(x)=0$ on $(\alpha, \beta)$ (C) there exists odd number of roots of the equation $\phi^{\prime}(x)$. $\mathrm{f}(\mathrm{x})+\phi "(\mathrm{x}) \cdot \mathrm{f}(\mathrm{x})=0$ on $(\alpha, \beta)$ (D) None of these

If $\mathrm{f}(\mathrm{x})$ is a differentiable function and $\phi(\mathrm{x})$ is twice differentiable function and $\alpha$ and $\beta$ are roots of the equation $f(x)=0$ and $\phi^{\prime}(x)=0$ respectively, then which of the following statement is true ? $(\alpha<\beta)$
(A) there exists exactly one root of the equation $\phi^{\prime}(x)$. $\mathrm{f}(\mathrm{x})+\phi^{\prime \prime}(\mathrm{x}) \cdot \mathrm{f}(\mathrm{x})=0$ on $(\alpha, \beta)$
(B) there exists atleast one root of the equation $\phi^{\prime}(x)$. $f(x)+\phi^{\prime \prime}(x) \cdot f(x)=0$ on $(\alpha, \beta)$
(C) there exists odd number of roots of the equation $\phi^{\prime}(x)$. $\mathrm{f}(\mathrm{x})+\phi "(\mathrm{x}) \cdot \mathrm{f}(\mathrm{x})=0$ on $(\alpha, \beta)$
(D) None of these

Key Concepts

Mean Value Theorem

The Mean Value Theorem extends the idea of Rolle’s Theorem by stating that a differentiable function on a closed interval attains a derivative equal to the average rate of change over that interval. This theorem is central to understanding the behavior of functions between two points and is frequently employed to demonstrate the existence of particular points where the function's instantaneous rate of change meets certain criteria.

Transformation Techniques Using Integrating Factors

Transformation techniques, such as multiplying a function by an integrating factor (often an exponential function), are used to simplify differential equations or expressions involving derivatives. By rewriting the equation in a form that represents the derivative of a product, these techniques allow the application of the aforementioned theorems (like Rolle's or the Mean Value Theorem) to deduce the existence and number of roots in an interval.

Rolle's Theorem

Rolle's Theorem is a fundamental result in calculus stating that if a function is continuous on a closed interval, differentiable on the open interval, and takes equal values at the endpoints, then there exists at least one point within the interval where the first derivative is zero. It is often used to establish the existence (and sometimes the uniqueness) of stationary points between given roots of a function.

Higher Order Differentiability

A function being twice differentiable means that not only does the first derivative exist, but the second derivative does as well. This second derivative gives information about the concavity and curvature of the function, which is important when analyzing more complex relationships between a function and its derivatives.

Differentiability

Differentiability refers to the property of a function where a derivative exists at every point in its domain. It reflects the idea that the function has a well?defined slope at every point, allowing us to study its instantaneous rate of change and apply various fundamental theorems in calculus.

Please give Ace some feedback