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utexas.instructure.com Files Try Ace AI STEM Tutor by Numerade Page 7 of 12 Explanation: When \( y=3+5 x \), \[ P=x y=x(3+5 x)=3 x+5 x^{2} \] Now the minimum value of \( P \) will occur at a critical point of \( P \), i.e., at the solution, \( x_{0} \), of \[ P^{\prime}(x)=3+10 x=0 \] in other words when \[ x_{0}=-\frac{3}{10} . \] 1. \( g(x)=2 \sqrt{x}\left(4 x^{2}+x-3\right)+C \) Since \[ P^{\prime \prime}\left(x_{0}\right)=10>0 \] we thus see that \[ \text { minimum value }=-\frac{9}{20} \] \( 010 \quad 10.0 \) points Find the most general antiderivative, \( F \), of 2. \( g(x)=2 \sqrt{x}\left(\frac{4}{5} x^{2}+\frac{1}{3} x-3\right)+C \) 3. \( g(x)=\sqrt{x}\left(4 x^{2}+x+3\right)+C \) 4. \( g(x)=\sqrt{x}\left(\frac{4}{5} x^{2}+\frac{1}{3} x+3\right)+C \) 5. \( g(x)=2 \sqrt{x}\left(4 x^{2}+x+3\right)+C \) the function \[ f(x)=6 x^{2}-10 x+6 \] 6. \( g(x)=2 \sqrt{x}\left(\frac{4}{5} x^{2}+\frac{1}{3} x+3\right)+C \) correct 1. \( F(x)=6 x^{3}-10 x^{2}+6 x+C \) 2. \( F(x)=2 x^{3}+5 x^{2}+6 x+C \) Explanation: After division 3. \( F(x)=2 x^{3}-5 x^{2}+6 x+C \) correct \[ g^{\prime}(x)=4 x^{3 / 2}+x^{1 / 2}+3 x^{-1 / 2} \]

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Try Ace AI STEM Tutor by Numerade
Page
7
of 12

Explanation:
When \( y=3+5 x \),
\[
P=x y=x(3+5 x)=3 x+5 x^{2}
\]

Now the minimum value of \( P \) will occur at a critical point of \( P \), i.e., at the solution, \( x_{0} \), of
\[
P^{\prime}(x)=3+10 x=0
\]
in other words when
\[
x_{0}=-\frac{3}{10} .
\]
1. \( g(x)=2 \sqrt{x}\left(4 x^{2}+x-3\right)+C \)

Since
\[
P^{\prime \prime}\left(x_{0}\right)=10>0
\]
we thus see that
\[
\text { minimum value }=-\frac{9}{20}
\]
\( 010 \quad 10.0 \) points
Find the most general antiderivative, \( F \), of
2. \( g(x)=2 \sqrt{x}\left(\frac{4}{5} x^{2}+\frac{1}{3} x-3\right)+C \)
3. \( g(x)=\sqrt{x}\left(4 x^{2}+x+3\right)+C \)
4. \( g(x)=\sqrt{x}\left(\frac{4}{5} x^{2}+\frac{1}{3} x+3\right)+C \)
5. \( g(x)=2 \sqrt{x}\left(4 x^{2}+x+3\right)+C \) the function
\[
f(x)=6 x^{2}-10 x+6
\]
6. \( g(x)=2 \sqrt{x}\left(\frac{4}{5} x^{2}+\frac{1}{3} x+3\right)+C \) correct
1. \( F(x)=6 x^{3}-10 x^{2}+6 x+C \)
2. \( F(x)=2 x^{3}+5 x^{2}+6 x+C \)

Explanation:
After division
3. \( F(x)=2 x^{3}-5 x^{2}+6 x+C \) correct
\[
g^{\prime}(x)=4 x^{3 / 2}+x^{1 / 2}+3 x^{-1 / 2}
\]

utexas.instructure.com
Files
Try Ace AI STEM Tutor by Numerade
Page
7
of 12

Explanation:
When y=3+5 x,

P=x y=x(3+5 x)=3 x+5 x^2

Now the minimum value of P will occur at a critical point of P, i.e., at the solution, x0, of

P^'(x)=3+10 x=0

in other words when

x0=-(3)/(10) .

1. g(x)=2 √(x)(4 x^2+x-3)+C

Since

P^''(x0)=10>0

we thus see that

minimum value =-(9)/(20)

010 10.0 points
Find the most general antiderivative, F, of
2. g(x)=2 √(x)((4)/(5) x^2+(1)/(3) x-3)+C
3. g(x)=√(x)(4 x^2+x+3)+C
4. g(x)=√(x)((4)/(5) x^2+(1)/(3) x+3)+C
5. g(x)=2 √(x)(4 x^2+x+3)+C the function

f(x)=6 x^2-10 x+6

6. g(x)=2 √(x)((4)/(5) x^2+(1)/(3) x+3)+C correct
1. F(x)=6 x^3-10 x^2+6 x+C
2. F(x)=2 x^3+5 x^2+6 x+C

Explanation:
After division
3. F(x)=2 x^3-5 x^2+6 x+C correct

g^'(x)=4 x^3 / 2+x^1 / 2+3 x^-1 / 2

Added by Lisa B.

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