Resources

Resources

Bytelearn - cat image with glasses

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

Найти точки экстремума и иятервалы монотонкости функции: $y=\sqrt{3 x^{2}+x^{3}}$

View step-by-step help

View step-by-step help

Compare linear, exponential, and quadratic growth

Full solution

Q. Найти точки экстремума и иятервалы монотонкости функции:

y=\sqrt{3 x^{2}+x^{3}}

Find Derivative: To find the points of extremum and intervals of monotonicity, we first need to find the derivative of the function. $\newline$ $y=\sqrt{3 x^{2}+x^{3}}$ $\newline$ Let's find the derivative using the chain rule. $\newline$ $y' = \frac{1}{2\sqrt{3 x^{2}+x^{3}}} \cdot (6x + 3x^2)$ $\newline$ Simplify the derivative. $\newline$ $y' = \frac{3x(2+x)}{2\sqrt{3 x^{2}+x^{3}}}$
Find Critical Points: Next, we need to find the critical points by setting the derivative equal to zero and solving for x. $\newline$ $\frac{3x(2+x)}{2\sqrt{3 x^{2}+x^{3}}} = 0$ $\newline$ This equation is zero when the numerator is zero, so we set $3x(2+x) = 0$ and solve for x. $\newline$ $3x = 0 \quad \text{or} \quad 2+x = 0$ $\newline$ $x = 0 \quad \text{or} \quad x = -2$
Consider Domain: We must also consider the domain of the original function, as the derivative must exist within this domain. The function $y=\sqrt{3 x^{2}+x^{3}}$ is defined for all x where $3 x^{2}+x^{3} \geq 0$ . Factoring out $x^2$ , we get $x^2(3+x) \geq 0$ . This inequality holds for $x \leq -3$ and $x \geq 0$ . Therefore, the critical point at $x = -2$ is not in the domain of the function, and we only consider $x = 0$ .
Test Intervals: Now we need to test the intervals around the critical point $x = 0$ to determine the intervals of monotonicity. We choose test points $x = -1$ (which is not in the domain, so we ignore it) and $x = 1$ . $\newline$ For $x = 1$ : $\newline$ $y' = \frac{3(1)(2+1)}{2\sqrt{3(1)^{2}+(1)^{3}}} = \frac{9}{2\sqrt{6}} > 0$ $\newline$ Since the derivative is positive, the function is increasing on the interval $(0, \infty)$ .
Identify Minimum Point: For $x = -3$ (which is at the boundary of the domain), we notice that the function $y=\sqrt{3 x^{2}+x^{3}}$ simplifies to $y=\sqrt{x^2(3+x)}$ , which is zero at $x = -3$ . Since the function is zero and then starts increasing after $x = 0$ , we can say that $x = -3$ is a minimum point.
Summary: To summarize, the function has a minimum point at $x = -3$ and is increasing on the interval $(0, \infty)$ . There are no maximum points since the function does not decrease after any point.

More problems from Compare linear, exponential, and quadratic growth

Question

Jill bought a used motorcycle from a seller online for $1,200. The seller will charge her $5 a day to store the motorcycle at his house until she is able to pick it up. You can use a function to describe the total amount of money Jill will owe the seller if she waits

x

days to pick up the motorcycle.

\newline

Write an equation for the function. If it is linear, write it in the form

g(x) = mx + b

. If it is exponential, write it in the form

g(x) = a(b)^x

\newline

g(x) = \underline{\hspace{3em}}

Posted 2 months ago

Question

f(t)=-2t-7

change over the interval from

t=-7

t=-6

\newline

\newline

\left[\text{f(t) decreases by } 2\right]

\newline

\left[\text{f(t) increases by } 2\right]

\newline

\left[\text{f(t) increases by } 200\%\right]

\newline

\left[\text{f(t) decreases by a factor of } 2\right]

Posted 2 months ago

Question

g(t)=9^t

change over the interval from

t=1

t=3

\newline

\newline

\left[\text{g(t) decreases by a factor of } 81\right]

\newline

\left[\text{g(t) increases by a factor of } 81\right]

\newline

\left[\text{g(t) increases by } 18\%\right]

\newline

\left[\text{g(t) decreases by } 9\%\right]

Posted 2 months ago

Question

Both of these functions grow as

x

gets larger and larger. Which function eventually exceeds the other?

\newline

\newline

(A)f(x) = 8x + 3.3

\newline

(B)g(x) = 3.3^x - 5

Posted 3 months ago

Question

Both of these functions grow as

x

gets larger and larger. Which function eventually exceeds the other?

\newline

\newline

(A) \ f(x) = 2x + 9

\newline

(B)\ g(x) = 2^x

Posted 3 months ago

Question

f(x)

g(x)

are differentiable.

\newline

h(x)

h(x)=\frac{g(x)}{f(x)}

\newline

f(8)=1

f'(8)=-6

g(8)=8

g'(8)=-10

h'(8)

\newline

Simplify any fractions.

\newline

h'(8)=

Posted 2 months ago

Question

p(x)

q(x)

are differentiable.

\newline

r(x)

r(x)= \frac{p(x)}{q(x)}

\newline

p(3)= 2

p'(3)= 4

q(3)= 6

q'(3)= 9

r'(3)

\newline

Simplify any fractions.

\newline

r'(3)=

Posted 4 months ago

Question

u(x)

v(x)

are differentiable.

\newline

w(x)

w(x)= \frac{u(x)}{v(x)}

\newline

u(5)= 3

u'(5)= -2

v(5)= 7

v'(5)= 1

w'(5)

\newline

Simplify any fractions.

\newline

w'(5)=

Posted 4 months ago

Question

a(x)

b(x)

are differentiable.

\newline

c(x)

c(x)= \frac{a(x)}{b(x)}

\newline

a(2)= 4

a'(2)= 5

b(2)= 1

b'(2)= -3

c'(2)

\newline

Simplify any fractions.

\newline

c'(2)=

Posted 4 months ago

Question

m(x)

n(x)

are differentiable.

\newline

o(x)

o(x)= \frac{m(x)}{n(x)}

\newline

m(7)= 2

m'(7)= -1

n(7)= 4

n'(7)= 8

o'(7)

\newline

Simplify any fractions.

\newline

o'(7)=

Posted 4 months ago