Table of Contents

## What is the concavity if the second derivative is 0?

Since the second derivative is zero, the function is neither concave up nor concave down at x = 0. It could be still be a local maximum or a local minimum and it even could be an inflection point.

## What happens when the second derivative is 0?

The second derivative is zero (f (x) = 0): When the second derivative is zero, it corresponds to a possible inflection point. If the second derivative changes sign around the zero (from positive to negative, or negative to positive), then the point is an inflection point.

**How does the second derivative show concavity?**

The 2nd derivative is tells you how the slope of the tangent line to the graph is changing. If you’re moving from left to right, and the slope of the tangent line is increasing and the so the 2nd derivative is postitive, then the tangent line is rotating counter-clockwise. That makes the graph concave up.

**What happens when FC 0?**

3. If f ‘(c)=0 and f”(c)=0 then the test fails.

### What happens when the first and second derivative is 0?

When x is a critical point of f(x) and the second derivative of f(x) is zero, then we learn no new information about the point. The point x may be a local maximum or a local minimum, and the function may also be increasing or decreasing at that point.

### What does it mean when the first and second derivative is 0?

**What happens when f ‘( c )= 0?**

If f ‘(c)=0 and f”(c)>0 then f has a local minimum at x=c.

**What does a derivative of 0 mean?**

The derivative of a function, f(x) being zero at a point, p means that p is a stationary point. That is, not “moving” (rate of change is 0).

## Can there be no concavity?

If the graph of a function is linear on some interval in its domain, its second derivative will be zero, and it is said to have no concavity on that interval.

## What does it mean when the derivative is zero?

Note: when the derivative curve is equal to zero, the original function must be at a critical point, that is, the curve is changing from increasing to decreasing or visa versa.

**How do you know if a function is concave up or down?**

The derivative of a function gives the slope.

- When the slope continually increases, the function is concave upward.
- When the slope continually decreases, the function is concave downward.

**What does F ‘( c 0 imply about X C?**

f'(c) = 0, means c is a critical point to f(x). From the Second Derivative Test if f”(c)>0, then f(x) has a local minimum at x = c. From the Second Derivative Test if f”(x)<0, then f(x) has a local maximum at x = c.

### What does the 2nd derivative tell you?

The second derivative measures the instantaneous rate of change of the first derivative. The sign of the second derivative tells us whether the slope of the tangent line to f is increasing or decreasing.

### What happens if the first derivative is 0?

The first derivative of a point is the slope of the tangent line at that point. When the slope of the tangent line is 0, the point is either a local minimum or a local maximum. Thus when the first derivative of a point is 0, the point is the location of a local minimum or maximum.

**What does it mean if f ‘( c )= 0?**

If f (c) = 0 and f (c) > 0 then f has a local minimum at c. • If f (c) = 0 and f (c) < 0 then f has a local maximum at c. x.

**What does second derivative tell you?**

Positive first derivative means an increasing function.

## What does first and second derivative mean?

– The first one is how position is changing over time. So, at periodic points, the position is measured. – The second one is the measure of speed (because we are not considering direction), how much the position changes per unit time period. This is the first derivative. – The third one is the acceleration, how much the velocity changes per unit time period.

## What does second derivative at a point represent?

The second derivative measures the instantaneous rate of change of the first derivative. The sign of the second derivative tells us whether the slope of the tangent line to (f) is increasing or decreasing. A differentiable function is concave up whenever its first derivative is increasing (or equivalently whenever its second derivative is positive), and concave down whenever its first derivative is decreasing (or equivalently whenever its second derivative is negative).

**What is the second derivative used for?**

The second derivative may be used to determine local extrema of a function under certain conditions. If a function has a critical point for which f′ (x) = 0 and the second derivative is positive at this point, then f has a local minimum here.