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

(i) f(x) = (2x − 1)2 + 3        (ii) f(x) = 9x2 + 12x + 2

(iii) f(x) = −(x − 1)2 + 10    (iv) g(x) = x3 + 1

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(i) The given function is f(x) = (2x − 1)2 + 3. It can be observed that (2x − 1)2 ≥ 0 for every x ∈ R. Therefore, f(x) = (2x − 1)2 + 3 ≥ 3 for every x ∈ R. The minimum value of f is attained when 2x − 1 = 0. 2x − 1 = 0 ⇒ ∴Minimum value of f ==...
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(i) The given function is f(x) = (2x − 1)2 + 3.

It can be observed that (2x − 1)2 ≥ 0 for every xR.

Therefore, f(x) = (2x − 1)2 + 3 ≥ 3 for every xR.

The minimum value of f is attained when 2x − 1 = 0.

2x − 1 = 0 ⇒

∴Minimum value of f == 3

Hence, function f does not have a maximum value.

(ii) The given function is f(x) = 9x2 + 12x + 2 = (3x + 2)2 − 2.

It can be observed that (3x + 2)2 ≥ 0 for every xR.

Therefore, f(x) = (3x + 2)2 − 2 ≥ −2 for every xR.

The minimum value of f is attained when 3x + 2 = 0.

3x + 2 = 0 ⇒

∴Minimum value of f =

Hence, function f does not have a maximum value.

(iii) The given function is f(x) = − (x − 1)2 + 10.

It can be observed that (x − 1)2 ≥ 0 for every xR.

Therefore, f(x) = − (x − 1)2 + 10 ≤ 10 for every xR.

The maximum value of f is attained when (x − 1) = 0.

(x − 1) = 0 ⇒ x = 1

∴Maximum value of f = f(1) = − (1 − 1)2 + 10 = 10

Hence, function f does not have a minimum value.

(iv) The given function is g(x) = x3 + 1.

Hence, function g neither has a maximum value nor a minimum value.

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Related Questions

For all real values of x, the minimum value of  is

(A) 0 (B) 1

(C) 3 (D)

Let ∴By second derivative test, f is the minimum at x = 1 and the minimum value is given by . The correct answer is D.
Chiranjeevi
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