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Evgesh-ka [11]
3 years ago
7

Find the x-coordinates of any relative extrema and inflection point(s) for the function f(x) = 6x(1/3) + 3x(4/3). You must justi

fy your answer using an analysis of f ′(x) and f ′′(x)
Mathematics
1 answer:
stealth61 [152]3 years ago
8 0
Applying our power rule gets us our first derivative,

\rm f'(x)=6\frac13x^{-2/3}+3\cdot\frac43x^{1/3}

simplifying a little bit,

\rm f'(x)=2x^{-2/3}+4x^{1/3}

looking for critical points,

\rm 0=2x^{-2/3}+4x^{1/3}

We can apply more factoring.
I hope this next step isn't too confusing.
We want to factor out the smallest power of x from both terms,
and also the 2 from each.

0=2x^{-2/3}\left(1+2x\right)

When you divide x^(-2/3) out of x^(1/3),
it leaves you with x^(3/3) or simply x.

Then apply your Zero-Factor Property,

\rm 0=2x^{-2/3}\qquad\qquad\qquad 0=(1+2x)

and solve for x in each case to find your critical points.

Apply your First Derivative Test to further classify these points. You should end up finding that x=-1/2 is an relative extreme value, while x=0 is not.

Let's come back to this,

\rm f'(x)=2x^{-2/3}+4x^{1/3}

and take our second derivative.

\rm f''(x)=-\frac43x^{-5/3}+\frac43x^{-2/3}

Looking for inflection points,

\rm 0=-\frac43x^{-5/3}+\frac43x^{-2/3}

Again, pulling out the smaller power of x, and fractional part,

\rm 0=-\frac43x^{-5/3}\left(1-x\right)

And again, apply your Zero-Factor Property, setting each factor to zero and solving for x in each case. You should find that x=0 and x=1 are possible inflection points.

Applying your Second Derivative Test should verify that both points are in fact inflection points, locations where the function changes concavity.
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NEED HELP NOW!!!!
Dovator [93]

Answer:

f(x) = -3x --->#6

f(x) = |x-1|+3 --->#5

f(x) = √(x+3) --->#3

1/2x² --->#1

f(x) = (x+1)²-3 --->#4

4|x|--->#2

Step-by-step explanation:

Recall for transformations:

  • Adding a number outside the function moves it up
  • Subtracting a number outside the function moves it down
  • Adding inside the function moves it to the left
  • Subtracting inside the function moves it to the right
  • Multiplying to the function by a number less than 1 compresses
  • Multiplying to a function by a number greater than 1 stretched it
  • Multiplying by a negative flips the graph

f(x) = -3x

This is multiplication by a number greater than 1 and a negative so this stretches and flip. This is #6, a reflection.

f(x) = |x-1|+3

Subtraction inside the function shifts it to the right 1 and addition outside shifts it up 3. This is #5.

f(x) = √(x+3)

Addition inside the function shifts it to the left 3. This is #3

1/2x²

Multiplication by 1/2 which is less than 1 compresses it. This is #1.

f(x) = (x+1)²-3

Addition inside the function shifts the function to the left once. This is #4.

4|x|

Multiplying by 4, a number greater than 1, stretches it. This is #2.


5 0
3 years ago
Find the estimates of x^2-x-2=3
Julli [10]

Answer:

x=\frac{1+\sqrt{21}}{2}\approx 2.79\\or\ x=\frac{1-\sqrt{21}}{2}\approx -1.79

Step-by-step explanation:

Given

Quadratic equation is

x^2-x-2=3\\x^2-x-5=0

Roots of the quadratic equation ax^2+bx+c=0 are

x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\\

So, for the given equation roots are

x=\frac{1\pm\sqrt{(-1)^2-4(1)(-5)}}{2\cdot 1}\\x=\frac{1\pm\sqrt{21}}{2}\\so, x=\frac{1+\sqrt{21}}{2}\approx 2.79\\or\ x=\frac{1-\sqrt{21}}{2}\approx -1.79

5 0
3 years ago
Paul opens a savings account with $350. He saves $150 per month. Assume that he does not withdraw money or make any additional d
Verizon [17]
I believe the equation is y=350+150x with y being the total amount of money in his account and x being the number of months..Paul starts with $350, which is the y-intercept (starting value) of the equation then the slope is 150 because his total savings increases by $150 for every month he saves without making any withdrawals. I apologize if I'm wrong but I hope this helps.
8 0
3 years ago
Mark wants to be the best football on his team. So he hits the weight room. He started lifting at 135lbs for his bench and every
balu736 [363]

Answer:

One week and around two days.

Step-by-step explanation:

7 0
3 years ago
Read 2 more answers
Harvard University accepts 6 students for every 100 applicants. How many students will be accepted in 850 applicants apply for a
LenKa [72]

Answer:

51 students.

Step-by-step explanation:

First you need to establish a rate 6/100+3/50

8*100=800   6*8=48

3/50

850-800=50   3/50

48+3=51



5 0
3 years ago
Read 2 more answers
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