Given $f(x)=|\sin 4 x+3|$ on $R$ We know that $-1 \leq \sin 4 x \leq 1$ $ \begin{array}{l} \Rightarrow 2 \leq \sin 4 x+3 \leq 4 \\ \Rightarrow 2 \leq|\sin 4 x+3| \leq 4 \end{array} $ Hence, the...

Given $f(x)=\sin 2 x+5$ on $R$ We know that $-1 \leq \sin 2 x \leq 1$ $ \begin{array}{l} \Rightarrow-1+5 \leq \sin 2 x+5 \leq 1+5 \\ \Rightarrow 4 \leq \sin 2 x+5 \leq 6 \end{array} $ Hence, the...

Given $f(x)=|x+2|$ on $R$ $\Rightarrow f(x) \geq 0$ for all $x \in R$ So, the minimum value of $f(x)$ is 0, which attains at $x=-2$ Hence, $f(x)=|x+2|$ does not have the maximum value.

Given $f(x)=-(x-1)^{2}+2$ It can be observed that $(x-1)^{2} \geq 0$ for every $x \in R$ Therefore, $f(x)=-(x-1)^{2}+2 \leq 2$ for every $x \in R$ The maximum value of $f$ is attained when $(x-1)=0$...

Given $f(x)=4 x^{2}-4 x+4$ on $R$ $=4 x^{2}-4 x+1+3$ By grouping the above equation we get, $ =(2 x-1)^{2}+3 $ Since, $(2 x-1)^{2} \geq 0$ $ \begin{array}{l} =(2 x-1)^{2}+3 \geq 3 \\ =f(x) \geq f(1...