Derivatives of order two or more are called higher derivatives and are represented by the following notation:
The last one is read as “the nth derivative of f with respect to x.”
The definition is given as follows by induction:
Exercise 1. Compute the second derivative for the following function:
Solution. First we compute the first derivative and then the second derivative for the assigned function:
Exercise 2. Compute the second derivative for the following function:
Solution. We compute the first derivative and then the second derivative for the assigned function:
Exercise 3. Compute the second derivative for the following function:
Solution. We compute the first derivative and then the second derivative for the assigned function:
Exercise 4. Compute the second derivative for the following function:
Solution. We compute before the first derivative and then the second derivative for the assigned function:
Definition. A function f with domain D is said to have an absolute maximum at c if f(x) ≤ f(c) for all x D. The number f(c) is called the absolute maximum of f on D. The function f is said to have a local maximum (or relative maximum) at c if there is some open interval (a, b) containing c and f(c) is the absolute maximum of f on (a, b).
Definition. A function f with domain D is said to have an absolute minimum at c if f(x) ≥ f(c) for all x D. The number f(c) is called the absolute minimum of f on D. The function f is said to have a local minimum (or relative minimum) at c if there is some open interval (a, b) containing c and f(c) is the absolute minimum of f on (a, b).
Definition. An absolute maximum or absolute minimum of f is called an absolute extremum of f. A local maximum or minimum of f is called a local extremum of f.
Theorem 1. (Extreme Value Theorem). If a function f is continuous on a closed and bounded interval [a,b], then there exist two points, c1 and c2, in [a,b] such that f(c1) is the absolute minimum of f on [a,b] and f(c2) is the absolute maximum of f on [a,b].
Theorem 2. If f is defined on an open interval (a,b) containing c, f(c) is a local extremum of f and f’(c) exists, then f’(c) = 0.
Definition. If f is differentiable at c and f’(c) = 0, then we call c a critical point or stationary point of f.
Definition. A function f is said to be increasing on an open interval (a,b) if f(x1)< f(x2) for all x1 and x2 in (a,b) such that x1< x2. The function f is said to be decreasing on (a,b) if f(x1) > f(x2) for all x1 and x2 in (a,b) such that x1< x2. The function f is said to be non-decreasing on (a,b) if f(x1) ≤ f(x2) for all x1 and x2 in (a,b) such that x1< x2. The function f is said to be non-increasing on (a,b) if f(x1)≥ f(x2) for all x1 and x2 in (a,b) such that x1< x2.
Theorem 3. Suppose that two functions f and g are continuous on a closed and bounded interval [a, b] and are differentiable on the open interval (a,b). Then the following statements are true:
(i) If f’(x) > 0 for each x in (a,b), then f is increasing on (a,b).
(ii) If f’(x) < 0 for each x in (a,b), then f is decreasing on (a, b).
(iii) If f’(x) ≥ 0 for each x in (a b), then f is non-decreasing on (a,b).
(iv) If f’(x) ≤ 0 for each x in (a,b), then f is non-increasing on (a,b).
(v) If f’(x) = 0 for each x in (a, b), then f is constant on (a, b).
Theorem 4. (First Derivative Test for Extremum). Let f be continuous on an open interval (a, b) and a < c < b.
(i) If f’(x) > 0 on (a,c) and f’(x) < 0 on (c,b), then f(c) is a local maximum of f on (a,b).
(ii) If f’(x) < 0 on (a,c) and f’(x) > 0 on (c,b), then f(c) is a local minimum of f on (a,b).
Exercise 5. Take the function
and locate the intervals where the graph of f is increasing or decreasing, the local extrema and absolute extrema on [-3, 3].
Solution. The function f(x) is defined for all real numbers. Hence, its domain is: x
[-3, 3]. Now we compute the first derivative to find stationary points and then apply the first derivative test for extrema:
To apply the first derivative test for extrema, we solve the following inequality:
and evaluate the sign of f’.
The function f is decreasing on and is increasing on
Then, applying the first derivative test for extrema, we say:
Now we evaluate the function f at the end points of the interval [-3, 3]:
Exercise 6. Take the function and locate the intervals where the graph of f is increasing or decreasing, the local extrema and absolute extrema on function’s domain.
Solution. The function f(x) is defined for all real number. Hence, its domain is: x
] – ∞, + ∞ [. Now we compute the first derivative to find stationary points and then apply the first derivative test for extrema:
So f’(x) = 0 when 1 – x2 = 0. Thus the function f has two critical points:
Now, to apply the first derivative test for extrema, we solve the following inequality:
and evaluate the sign of f’.
Now, applying the first derivative test for extrema, we say:
Moreover, we study the behavior of the function f at the end points of its domain ] -∞, + ∞ [:
Showing that the x-axis is a horizontal asymptote for the graph of f.
Now, putting these observations together, we can say that:
Exercise 7. Take the function and locate the intervals where the graph of f is increasing or decreasing, the local extrema and absolute extrema on function’s domain.
1. Functions and their graph. One–one and onto functions. Composite and inverse functions.
2. Linear inequalities. Systems of linear inequalities
3. Polynomial inequalities. Rational inequalities. Absolute-value inequalities
6. Differentiation. Derivative rules, the chain rule.
7. Application of differentiation: L'Hospital's Rule
8. Vertical, Horizontal and Slant asymptotes
9. Higher Order Derivatives. Applications of differentiation: local and absolute extremes of a function
10. Curve Sketching