Find the maximum and minimum values, if any, of the following functions given by(i) `f(x)=(2x-1)^2+3` (ii) `f(x)=9x^2+12 x+2`(iii) `f(x)=-(x-1)^2+10` (iv) `g(x)=x^3+1`

Explanation

Let's solve the given functions step by step to find their maximum and minimum values.

### (i) \( f(x) = (2x - 1)^2 + 3 \)

1. **Identify the nature of the function**: The term \( (2x - 1)^2 \) is a square term, which is always non-negative. Therefore, \( (2x - 1)^2 \geq 0 \).

2. **Determine the minimum value**: Since \( (2x - 1)^2 \) is always greater than or equal to 0, the minimum value of \( f(x) \) occurs when \( (2x - 1)^2 = 0 \).
- Setting \( (2x - 1)^2 = 0 \) gives \( 2x - 1 = 0 \) or \( x = \frac{1}{2} \).
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