Given the piecewise function below, select all of the statements that are true. f(x)= left{-x + 1, x < 0right} left{-2, x = 0right} left{x^{2} -1, x > 0right} A. f(4)=7 B. f(-1)=2 C. f(1)=0 D. f(-2)=0

Find the value of x that satisfies the inequality.
${\displaystyle \left|\frac{x-1}{2}\right|\le 1}$

Absolute value inequalities
9-3|2x-2|<-3

Simplify $$-14\sqrt{175}$$.

Calculate all the values of x such that $$-16x^2+96x+2560=0$$.

Prove $0$ is an exponentially stable equilibrium of the system $x^{\prime }=f(x)+g(x)$ if $f(0)=g(0)=0$
Besides the conditions in the title, we have:
1. $0$ is an exponential equilibrium of the system $y^{\prime }=f(y)$
2. $|g(x)|\le \mu |x|,\mathrm{\forall }x\in {\mathbb{R}}^n$
3. $\mu$ is sufficiently small!
What I have tried so far is that:
1. $||y(0)||<{\delta }_1\Rightarrow ||y(t)||<{\alpha }_1||y(0)||e^{-{\beta }_{1}t},t\ge 0$
2. We want to prove: $\mathrm{\exists }\delta ,\alpha ,\beta >0,$ such that
$||x(0)||<\delta \Rightarrow ||x(t)||<\alpha ||x(0)||e^{-\beta t},t\ge 0$
Considering the following system:
1. $x_1^{\prime }(t)=f(x_1)$
2. $x_2^{\prime }(t)=g(x_2)$
Then we will have:
1. $\mathrm{\exists }{\alpha }_1,{\beta }_1,\delta >0$ such that
$||x_1(0)||<{\delta }_1\Rightarrow ||x_1(t)||<{\alpha }_1||x_1(0)||e^{-{\beta }_{1}t},t\ge 0$
2. $x_2^{\prime }(t)\le \mu |x_2(t)|$, and by Gronwall inequality $x_2(t)\le x_2(0)e^{\mu t}$
I have a sense that it might be somewhat close to what we want to get. And I am stuck at this stage.

Solve the equation ${\displaystyle \sin x+\cos x=k\sin x\cos x}$ for real x, where k is a real constant.

Determine whether the sequence is arithmetic, geometric, or neither. If the sequence is arithmetic or geometric, calculate its common difference or ratio, respectively. 5, 9, 13, 17,