State the conditions for maxima, minima and for point of inflexion.

In calculus, the conditions for identifying maxima, minima, and points of inflection for a function \( f(x) \) involve analyzing its first and second derivatives. Here’s a summary of these conditions:

### 1. **Conditions for a Maximum**:
A point \( x = c \) is a **local maximum** of \( f(x) \) if:
- **First derivative test**: \( f'(c) = 0 \) and the sign of \( f'(x) \) changes from positive to negative as \( x \) passes through \( c \).
- **Second derivative test**: \( f'(c) = 0 \) and \( f''(c) < 0 \). This indicates the function is concave down at \( x = c \), meaning it has a peak.

### 2. **Conditions for a Minimum**:
A point \( x = c \) is a **local minimum** of \( f(x) \) if:
- **First derivative test**: \( f'(c) = 0 \) and the sign of \( f'(x) \) changes from negative to positive as \( x \) passes through \( c \).
- **Second derivative test**: \( f'(c) = 0 \) and \( f''(c) > 0 \). This indicates the function is concave up at \( x = c \), meaning it has a trough.

### 3. **Conditions for a Point of Inflection**:
A point \( x = c \) is a **point of inflection** of \( f(x) \) if:
- The sign of the second derivative \( f''(x) \) changes as \( x \) passes through \( c \). This means the concavity of the function changes from concave up to concave down, or vice versa.
- \( f''(c) = 0 \) is often a necessary condition for a point of inflection, but not sufficient on its own; the change in concavity must be verified by the sign change in \( f''(x) \).

### Summary:
- **Maxima**: \( f'(c) = 0 \) and \( f''(c) < 0 \) (concave down).
- **Minima**: \( f'(c) = 0 \) and \( f''(c) > 0 \) (concave up).
- **Inflection**: \( f''(c) = 0 \) and \( f''(x) \) changes sign around \( c \).

Would you like to go into further details or examples for these concepts?

Here are some related questions:

1. How do you determine the global maximum or minimum of a function on a closed interval?
2. What is the difference between a local and an absolute maximum?
3. Can a point of inflection occur at a point where \( f'(c) \neq 0 \)?
4. How does the first derivative test help in identifying local extrema?
5. What role does concavity play in understanding the behavior of a function?

**Tip**: For complex functions, always verify the sign change around critical points to confirm maxima, minima, or inflection points.

Understand the conditions for maxima, minima, and points of inflection in calculus. Learn about the first and second derivative tests and how they determine local extrema and changes in concavity. Suitable for undergraduate students and those preparing for calculus exams.

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