Answer to Question #344749 in Algebra for Geoffroy yawogan

Question

1. Find the domain of the function 𝑓(𝑥)=ln(−2𝑥2−𝑥−6)+√𝑥2−1.

 

2. Find the inverse function of the function 𝑓(𝑥)=𝑥2−4𝑥+5,𝑥∈〈3,4⟩. Find the domain and the range of the inverse function.

 

3. Construct the tangent line to the graph of the function 𝑓(𝑥)=4𝑥⋅√𝑥−2⋅√𝑥 which is parallel to the line 𝑦=𝑥.

 

4. Find the maximal intervals of monotonicity of the function 𝑓(𝑥)=𝑒𝑥+3𝑥2+2𝑥+6.

 

5. Find the integral ∫6⋅𝑥3⋅𝑒𝑥2+2𝑑𝑥.

 

6. Find the general solution of the differential equation 𝑥2+1+𝑦′⋅cos(𝑦)=0. 

Answer 1

1.

a)

$f(x)=\ln(-2x^2-x-6)+\sqrt{x^2-1}$

$-2x^2-x-6>0$ $x^2-1\ge0$

$D=(-1)^2-4(-2)(-6)=-47<0$

Then $-2x^2-x-6<0, x\in \R$

$Domain:\{\}$

b)

$f(x)=\ln(-\dfrac{2}{x^2-x-6})+\sqrt{x^2-1}$

$x^2-x-6<0$ $x^2-1\ge0$

$(x+2)(x-3)<0$

$x\le -1\ or\ x\ge1$

Domain:$(-2, -1]\cup[1,3)$

2.

$𝑓(𝑥)=x^2−4x+5,x\in [3, 4]$

$x_v=-\dfrac{-4}{2(1)}=2$

The function $f$ increases on $(3, 4)$

$f(3)=(3)^2−4(3)+5=2$

$f(4)=(4)^2−4(4)+5=5$

Domain: $[3, 4]$

Range: $[2, 5]$

$y=x^2-4x+5, 3\le x\le4$

Change $x$ and $y$

$x=y^2-4y+5, 3\le y\le 4$

Solve for $y$

$y^2-4y+4=x-1$ $(y-2)^2=x-1$

Since $3\le y\le 4$

$y-2=\sqrt{x-1}$

$y=2+\sqrt{x-1}$

Then

$f^{-1}(x)=2+\sqrt{x-1}$

Domain: $[2, 5]$

Range: $[3, 4]$

3.

$f(x)=4x\sqrt{x}-2\sqrt{x}$

Domain: $[0, \infin)$

$f'(x)=4(\dfrac{3}{2})\sqrt{x}-\dfrac{2}{2\sqrt{x}}=\dfrac{6x-1}{\sqrt{x}}$

$slope=f'(x)=\dfrac{6x-1}{\sqrt{x}}=1$

$6x-\sqrt{x}-1=0$

$(3\sqrt{x}+1)(2\sqrt{x}-1)=0$

Since $\sqrt{x}\ge0,$ we take $2\sqrt{x}-1=0$

$\sqrt{x}=\dfrac{1}{2}$

$x=\dfrac{1}{4}$

$f(\dfrac{1}{4})=4(\dfrac{1}{4})\sqrt{\dfrac{1}{4}}-2\sqrt{\dfrac{1}{4}}=-\dfrac{1}{2}$

The tangent line to the graph is

$y+\dfrac{1}{2}=x-\dfrac{1}{4}$

$y=x-\dfrac{3}{4}$

4.

$f(x)=e^x+3x^2+2x+6$

Domain: $(-\infin, \infin)$

$f'(x)=e^x+6x+2$

$f'(x)=0=>e^x+6x+2=0$

$(e^x+6x+2)'=e^x+6>0, x\in \R$

The function $g(x)=e^x+6x+2$ is strictly increasing on $(-\infin, \infin).$

The equation $e^x+6x+2=0$ has the only solution $x\approx-0.4406.$

The function $f(x)$ is monotonic decreasing on $(-\infin, -0.4406).$

The function $f(x)$ is monotonic increasing on $( -0.4406, \infin).$

5.

$\int 6x^3e^{x^2+2}dx$

$t=x^2, dt=2xdx$

$\int 6x^3e^{x^2+2}dx=3e^2\int te^tdt$

$\int te^t dt$

$u=t, du=dt$

$dv=e^tdt, v=e^t$

$\int te^t dt=te^t -\int e^t dt=te^t-e^t+C_1$

$\int 6x^3e^{x^2+2}dx=3e^{x^2+2}(x^2-1)+C$

 6.

$x^2+1+y'\cos(y)=0$

$\cos(y)dy=-(x^2+1)dx$

Integrate

$\int \cos(y)dy=-\int (x^2+1)dx$

$\sin (y)=-\dfrac{x^3}{3}-x+C$

$\sin (y)+\dfrac{x^3}{3}+x=C$