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
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
Solution and Explanation
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.
3x + 2 = 0 ∴ x=\(-\frac{2}{3}\)
∴The minimum value of f = f(\(-\frac{2}{3}\))=(3(\(-\frac{2}{3}\))+2)2
Hence, function f does not have a maximum value.
The given function is f(x) =−(x−1)2+10.
It can be observed that (x − 1) 2 ≥ 0 for every x ∴ R.
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Concepts Used:
Maxima and Minima
What are Maxima and Minima of a Function?
The extrema of a function are very well known as Maxima and minima. Maxima is the maximum and minima is the minimum value of a function within the given set of ranges.
There are two types of maxima and minima that exist in a function, such as:
- Local Maxima and Minima
- Absolute or Global Maxima and Minima