Question 3. (35) (Answer this on a page marked Question 3) Consider the function y = f(x) whose graph is sketched below. Using only this sketch; (a) Where is the derivative of f equal to zero? (b) Where is the derivative of f positive? (c) Sketch the graph of the derivative of f.

Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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**Question 3. (35) (Answer this on a page marked Question 3)**

Consider the function \( y = f(x) \) whose graph is sketched below. Using only this sketch:

(a) Where is the derivative of \( f \) equal to zero?

(b) Where is the derivative of \( f \) positive?

(c) Sketch the graph of the derivative of \( f \).

**Graph Explanation:**
The graph provided shows the function \( y = f(x) \). It features the following characteristics:

- The graph has a peak (local maximum) at around \( x = -1 \).
- There is a trough (local minimum) at \( x = 1 \).
- The function appears to be increasing for \( x < -1 \) and \( x > 1.5 \).
- It decreases between \( x = -1 \) and \( x = 1 \). 

To answer (a), identify points where the slope of the tangent to the curve is horizontal (i.e., derivative is zero) at \( x = -1 \) and \( x = 1 \).

For (b), the derivative is positive wherever the function is increasing, which occurs when \( x < -1 \) and \( x > 1.5 \).

(c) To sketch the derivative, plot a graph that is zero at \( x = -1 \) and \( x = 1 \), positive for \( x < -1 \) and \( x > 1.5 \), and negative between \( x = -1 \) and \( x = 1 \). The derivative graph will likely show changes in direction at these critical points.
Transcribed Image Text:**Question 3. (35) (Answer this on a page marked Question 3)** Consider the function \( y = f(x) \) whose graph is sketched below. Using only this sketch: (a) Where is the derivative of \( f \) equal to zero? (b) Where is the derivative of \( f \) positive? (c) Sketch the graph of the derivative of \( f \). **Graph Explanation:** The graph provided shows the function \( y = f(x) \). It features the following characteristics: - The graph has a peak (local maximum) at around \( x = -1 \). - There is a trough (local minimum) at \( x = 1 \). - The function appears to be increasing for \( x < -1 \) and \( x > 1.5 \). - It decreases between \( x = -1 \) and \( x = 1 \). To answer (a), identify points where the slope of the tangent to the curve is horizontal (i.e., derivative is zero) at \( x = -1 \) and \( x = 1 \). For (b), the derivative is positive wherever the function is increasing, which occurs when \( x < -1 \) and \( x > 1.5 \). (c) To sketch the derivative, plot a graph that is zero at \( x = -1 \) and \( x = 1 \), positive for \( x < -1 \) and \( x > 1.5 \), and negative between \( x = -1 \) and \( x = 1 \). The derivative graph will likely show changes in direction at these critical points.
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