SECOND DERIVATIVE TEST FOR LOCAL EXTREMA

Problem 1 :

Find intervals of concavity and points of inflexion for the following functions:

(i) f (x) = x(x − 4)³

(ii) y = sin x + cos x, 0 < x < 2π

f(x) = 1/2 (e^x-e^-x)

Problem 2 :

Find the local extrema for the following functions using second derivative test :

(i) f(x) = −3x⁵+ 5x³

(ii) f(x) = x logx

(iii) f(x) = x²e^-2xSolution

Problem 3 :

For the function

f(x) = 4x³+3x²-6x+1

find the intervals of

(i) monotonicity

(ii) local extrema

(iii) intervals of concavity and

(iv) points of inflection. Solution

More worksheets on

Answers :

Problem 1 :

(i) Concave up on (-∞, 2) and (4, π).

Concave down on (2, 4).

point of inflection are (2, -16) and (4, 0).

(ii) Concave down on (0, 3π/4) and (7π/4, 2π).

Concave up on (3π/4, 7π/4).

Point of inflection are (3π/4, 0) and (7π/4, 0).

(iii) Concave up on (-∞, 0) and concave down on (0, ∞)

point of inflection is (0, 0).

Problem 2 :

(i) local maximum point is (1, 2) and local maximum is 2.

(ii) local minimum is -1/e

(iii) Local maximum = 1/e²and Local minimum = 0

Problem 3 :

Concave downward on (-∞, -1/4) and Concave upward on (-1/4, ∞). Point of inflection is (-1/4, 21/8)

Apart from the stuff given above, if you need any other stuff in math, please use our google custom search here.

Kindly mail your feedback to v4formath@gmail.com

We always appreciate your feedback.

Recent Articles