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 (ex-e-x)

Solution

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-2x               Solution

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 

Application of first derivatives

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/e2 and Local minimum  =  0

Problem 3 :

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

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