Let f(x) = 2 ^ x - x ^ 2 , x ∈ R . If m and n are respectively the...

Given,

Function $f\left(x\right) = {2}^{x} - {x}^{2}, x \in R$

Now, plotting the diagram of $y = {2}^{x} \text{ and } y = {x}^{2}$ we get,

Now, from above diagram total point of intersection are three

$∴ m = 3$

Now, finding ${f}^{'} ( x ) = {2}^{x}\ln 2 - 2x = 0$

${2}^{x}\ln 2 = 2x$

Now, again plotting the diagram of $y = {2}^{x}\ln 2 \text{ and } y = 2x$ we get,

So, from above diagram total point of intersection is two

$∴ n = 2$

Hence, $m + n = 5$