Question

f(x) = 4x² + 4 on R .

Answer 1

Given: f(x) = 4x² − 4x + 4

$$\Rightarrow$$ f(x) = (4x² − 4x + 1)+3

$$\Rightarrow$$ f(x) = (2x − 1)² + 3

Now,
(2x − 1)² $$\ge$$ 0 for all x $$\in$$ R $$\in$$

$$\Rightarrow$$ f(x) = (2x − 1)² + 3 $$\ge$$ 3 for all x $$\in$$ R

$$\Rightarrow$$ f(x) $$\ge$$ 3 for all x $$\in$$ R

The minimum value of f is attained when (x − 1) = 0.
(2x − 1) = 0
⇒ x = $$\frac{1}{2}$$

Thus, the minimum value of f (x) at x = $$\frac{1}{2}$$ is 3.

Since f(x) can be enlarged, the maximum value does not exist, which is evident in the graph also.

Hence, function f does not have a maximum value .