# The second derivative of function f.

### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

#### Solutions

Chapter 2, Problem 13RCC
To determine

## To define: The second derivative of function f.

Expert Solution

### Explanation of Solution

(1) The second derivative is derivative of the derivative function.

f(a)=limxaf(x)f(a)xaf(a)=limh0f(x+h)f(x)h

(2) The second derivative exhibits whether a point with zero is critical point.

Case (i): d2ydt2<0

The second derivative is negative for the maximum point. The slope of the curve is at first positive, then goes through zero to become negative.

Case (ii): d2ydt2>0

The second derivative is positive for the minimum point. The slope of the curve is at first negative, then goes through zero to become positive.

Case (iii): d2ydt2=0

The second derivative and first derivative both are zero for the inflexion point at same time. It represents a point where the curvature is changing its sense. Inflexion points are relatively rare.

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