Prove monotonic function on $\mathbb{R}$ under given condition

Question

Let $f(x)=e^{x}-x^2-ax$

(a) Prove when $a\leq 2-2ln(2)$ , $f(x)$ is monotonic function on $\mathbb{R},(a,+\infty)$

(b) Given when $x>0$, $f(x)\geq 1-x$ always true. Find the range of $a$.

This is a question from middle school exam paper on the topic related to monotonic function. Can anyone solve this? I was wondering why middle school exam contains such a weird question.

Answer 1

Hint:

Try to use $x=\ln(y)$.

Then you have $f(y) = y-\ln(y)^2-a\ln(y)$.

Then $f'(y)= 1-\frac{2}{y}\ln(y)-\frac{a}{y}$.