3. Suppose the function f has second derivative f"(x) = x2(x – 3)(x+2)*(? – 1). = T- (a) How many inflection points does f have? Where do they occur? 1 (b) Suppose f has horizontal tangent lines at = -5, 0, and 2. - Which correspond to local maximum? a local minimum? Which cannot be determined?
3. Suppose the function f has second derivative f"(x) = x2(x – 3)(x+2)*(? – 1). = T- (a) How many inflection points does f have? Where do they occur? 1 (b) Suppose f has horizontal tangent lines at = -5, 0, and 2. - Which correspond to local maximum? a local minimum? Which cannot be determined?
3. Suppose the function f has second derivative f"(x) = x2(x – 3)(x+2)*(? – 1). = T- (a) How many inflection points does f have? Where do they occur? 1 (b) Suppose f has horizontal tangent lines at = -5, 0, and 2. - Which correspond to local maximum? a local minimum? Which cannot be determined?
Hi, I have no clue to find local maximum and minimum from problem 4b. I tried to watch videos to get a clue on where to start. Ty
Transcribed Image Text:**Problem 3: Analysis of Function with Given Second Derivative**
Given:
The function \( f \) has the second derivative:
\[ f''(x) = x^2(x - 3)^4(x + 2)^3(x^2 - 1). \]
**(a) Inflection Points:**
How many inflection points does \( f \) have? Where do they occur?
**(b) Horizontal Tangent Lines:**
Suppose \( f \) has horizontal tangent lines at \( x = -5, 0, \frac{1}{2}, \) and \( 2 \).
- Which of these correspond to a local maximum?
- Which correspond to a local minimum?
- Which cannot be determined?
**Explanation:**
To solve for inflection points in part (a), observe where the second derivative changes sign, indicating a change in concavity.
For part (b), determine which of the given points correspond to local extrema by analyzing the behavior of the first derivative, particularly where it equals zero, and the second derivative for concavity.
Formula Formula A function f(x) attains a local maximum at x=a , if there exists a neighborhood (a−δ,a+δ) of a such that, f(x)<f(a), ∀ x∈(a−δ,a+δ),x≠a f(x)−f(a)<0, ∀ x∈(a−δ,a+δ),x≠a In such case, f(a) attains a local maximum value f(x) at x=a .
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![**Problem 3: Analysis of Function with Given Second Derivative**
Given:
The function \( f \) has the second derivative:
\[ f''(x) = x^2(x - 3)^4(x + 2)^3(x^2 - 1). \]
**(a) Inflection Points:**
How many inflection points does \( f \) have? Where do they occur?
**(b) Horizontal Tangent Lines:**
Suppose \( f \) has horizontal tangent lines at \( x = -5, 0, \frac{1}{2}, \) and \( 2 \).
- Which of these correspond to a local maximum?
- Which correspond to a local minimum?
- Which cannot be determined?
**Explanation:**
To solve for inflection points in part (a), observe where the second derivative changes sign, indicating a change in concavity.
For part (b), determine which of the given points correspond to local extrema by analyzing the behavior of the first derivative, particularly where it equals zero, and the second derivative for concavity.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6e1dfc2a-8d34-4823-9643-496b8fa276f1%2F11515367-827d-4de9-b4f2-651a848412a1%2Fysgzsn_processed.jpeg&w=3840&q=75)
