Question 42 of 82intermediate🔧 ApplyShort Answer4 marks
Find the absolute maximum value and the absolute minimum value of the following functions in the given intervals:
(i)
f(x) = x³, x ∈ [-2, 2]
Answer
Absolute maximum value = 8 at x = 2; Absolute minimum value = -8 at x = -2
(ii)
f(x) = sin x + cos x, x ∈ [0, π]
Answer
Absolute maximum value = at x = π/4; Absolute minimum value = -1 at x = π
(iii)
f(x) = 4x - (1/2)x², x ∈ [-2, 9/2]
Answer
Absolute maximum value = 8 at x = 4; Absolute minimum value = -10 at x = -2
(iv)
f(x) = (x-1)² + 3, x ∈ [-3, 1]
Answer
Absolute maximum value = 19 at x = -3; Absolute minimum value = 3 at x = 1
Explanation
To find absolute maximum and minimum values in a closed interval, we evaluate the function at all critical points (where f'(x) = 0 or f'(x) is undefined) and at the endpoints of the interval, then compare all values.