https://file.aiquickdraw.com/mathbot/file/1726597908969mwpej2ud.jpg

The image contains two mathematical exercises. Let's go through the questions from Exercise 9:

### Exercise 9 Details:
This exercise is about the function \( f \) defined as \( f(x) = ax + b + e^{-0.1x} \) where \( a \) and \( b \) are constants. The function's graph, labeled \( C_f \), and its tangent at a specific point are depicted.

#### Questions:
1. **Determine the value of \( b \)**:
   - The tangent line \( T \) to the curve at a given point passes through \( B(4;19) \).
   - Use the point \( (4, 19) \) and the derivative \( f'(x) \) to find \( b \).

2. **Equation of Tangent Line \( T \)**:
   - **Part a:** Find the equation of the tangent line \( T \).
   - **Part b:** Express \( f'(x) \) in terms of \( a \), \( d \), and \( e \), then find the general expression for \( x \).

3. **Determine the maximum of \( f(x) \) on \( \mathbb{R} \)**:
   - Analyze the first derivative to find critical points and determine if it's a maximum.

Let's start by calculating the value of \( b \) using the information given:

We know that the tangent line at point \( A \) (which needs the x-coordinate to be specified from the graph) has the point \( (4,19) \) and a slope given by the derivative at that x-coordinate.

Would you like me to calculate \( b \) using the point-slope form of the tangent line, and then proceed with the other parts? Or do you have specific coordinates for point \( A \) or further instructions?

We consider the function f defined and differentiable over ℝ by f(x) = ax + b + e^{-0.1x} where a and b are real constants. The curve representing the function f is given below, in an orthogonal coordinate system. We also show the tangent T to the curve Cf at point A(0;5). We admit that this tangent passes through point B(4;19).

1. Determine the value of b.

2. (a) Determine the equation of the line T. (b) Express, for any real x, f'(x) in terms of a, d, and e, then deduce that for any real x, f'(x) = 4x + 5 * e^{-0.1x}.

3. Determine the maximum of the function f on ℝ. Justify.

Learn how to determine the tangent line and find the maximum of the function f(x) = ax + b + e^{-0.1x). This problem involves differentiation and understanding the behavior of exponential functions. Suitable for advanced high school or university-level students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation