Question

Find the absolute maximum value and the absolute minimum value of the following functions in the given intervals:

(i) (ii)

(iii)

(iv)

Answer 1

(i) The given function is f(x) = x³.

Then, we evaluate the value of f at critical point x = 0 and at end points of the interval [−2, 2].

f(0) = 0

f(−2) = (−2)³ = −8

f(2) = (2)³ = 8

Hence, we can conclude that the absolute maximum value of f on [−2, 2] is 8 occurring at x = 2. Also, the absolute minimum value of f on [−2, 2] is −8 occurring at x = −2.

(ii) The given function is f(x) = sin x + cos x.

Then, we evaluate the value of f at critical point and at the end points of the interval [0, π].

Hence, we can conclude that the absolute maximum value of f on [0, π] is occurring atand the absolute minimum value of f on [0, π] is −1 occurring at x = π.

(iii) The given function is

Then, we evaluate the value of f at critical point x = 4 and at the end points of the interval.

Hence, we can conclude that the absolute maximum value of f onis 8 occurring at x = 4 and the absolute minimum value of f on is −10 occurring at x = −2.

(iv) The given function is

Now,

2(x − 1) = 0 ⇒ x = 1

Then, we evaluate the value of f at critical point x = 1 and at the end points of the interval [−3, 1].

Hence, we can conclude that the absolute maximum value of f on [−3, 1] is 19 occurring at x = −3 and the minimum value of f on [−3, 1] is 3 occurring at x = 1.