Which of following inequalities hold in interval 0 to pi/2

Question

i tried using calculator and i got 1,2,4 correct .But i am not sure about how to prove them

Answer 1

For C, I recognize the right side as the first 3 terms of the binomial theorem expansion of $\sqrt{1+x}$.

If it is squared, we get

$\begin{array}\\ (1+\frac{x}{2}-\frac{x^2}{8})^2 &=1+\frac{x^2}{4}+\frac{x^4}{64} +2\frac{x}{2}-2\frac{x^2}{8}-2\frac{x^3}{16}\\ &=1+x-\frac{x^3}{8}+\frac{x^4}{64}\\ &=1+x-\frac{x^3}{8}(1-\frac{x}{8})\\ &< 1+x\\ \end{array} $

Answer 2

D) is correct. Since the exponential function is monotone, it is equivalent to saying that the function $f(x) = 2 + x - e^{(1-x^2)/2}$ is positive on the interval $(0, \pi/2)$. We obviously have $f(0) > 0$ and $$ \frac{d}{dx}f(x) = 1 + xe^{\frac{1-x^2}{2}} > 0 \text{ for }x \in (0, \frac{\pi}{2}) $$ which shows the desired inequality.