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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)
Asked by Shabeena Last Modified
Answer 
Tasneem
(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 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).
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Doonu Singh
I am expert in electrical engineering . I have b tech and m tech from nit warangal total experi 10 y
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