Question

Find the maximum and minimum values, if any, of the following functions given by

(i) f(x) = |x + 2| − 1 (ii) g(x) = − |x + 1| + 3

(iii) h(x) = sin(2x) + 5 (iv) f(x) = |sin 4x + 3|

(v) h(x) = x + 1, x (−1, 1)

Answer 1

(i) f(x) =

We know that for every x ∈ R.

Therefore, f(x) = for every x ∈ R.

The minimum value of f is attained when.

∴Minimum value of f = f(−2) =

Hence, function f does not have a maximum value.

(ii) g(x) =

We know that for every x ∈ R.

Therefore, g(x) = for every x ∈ R.

The maximum value of g is attained when.

∴Maximum value of g = g(−1) =

Hence, function g does not have a minimum value.

(iii) h(x) = sin2x + 5

We know that − 1 ≤ sin 2x ≤ 1.

⇒ − 1 + 5 ≤ sin 2x + 5 ≤ 1 + 5

⇒ 4 ≤ sin 2x + 5 ≤ 6

Hence, the maximum and minimum values of h are 6 and 4 respectively.

(iv) f(x) =

We know that −1 ≤ sin 4x ≤ 1.

⇒ 2 ≤ sin 4x + 3 ≤ 4

⇒ 2 ≤≤ 4

Hence, the maximum and minimum values of f are 4 and 2 respectively.

(v) h(x) = x + 1, x ∈ (−1, 1)

Here, if a point x₀ is closest to −1, then we find $\frac{x_0}{2}+1>x_0+1$ for all x₀ ∈ (−1, 1).

Also, if x₁ is closest to 1, then for all x₁ ∈ (−1, 1).

Hence, function h(x) has neither maximum nor minimum value in (−1, 1).