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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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Ierofanga [76]

Answer:

           slope:      \bold{m=3}

      equation:     \bold{y-2=3(x-1)}

Step-by-step explanation:

m=\dfrac{y_2-y_1}{x_2-x_1}\\\\(1,\,2)\ \implies\ x_1=1\,,\ y_1=2\\(0,\,-1)\ \implies\ x_2=0\,,\ y_2=-1\\\\m=\dfrac{-1-2}{0-1}=\dfrac{-3}{-1}=3

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x_1=1\,,\ y_1=2\,,\ m=3\\\\equation:\\{}\qquad\qquad y-2=3(x-1)

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Triss [41]
The answer is -4, hope this helped
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3 years ago
Find the values of x and y
julsineya [31]

Answer:

y = 70 degrees

x = 11

Step-by-step explanation:

The side opposite x + 2 is equivalent so they will both be equal to 13.

To get the meaning of x, simply subtract 2 from 13 giving you x = 11.

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4 0
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What is 15% of 135, show your work
SIZIF [17.4K]

Answer:

20.25

Step-by-step explanation:

Percentage solution with steps:

Step 1: Our output value is 135.

Step 2: We represent the unknown value with $x$

.

Step 3: From step 1 above,$135=100\%$

.

Step 4: Similarly, $x=15.\%$

.

Step 5: This results in a pair of simple equations:

$135=100\%(1)$.

$x=15.\%(2)$

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Step 6: By dividing equation 1 by equation 2 and noting that both the RHS (right hand side) of both

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$\frac{135}{x}=\frac{100\%}{15.\%}$

Step 7: Again, the reciprocal of both sides gives

$\frac{x}{135}=\frac{15.}{100}$

$\Rightarrow x=20.25$Therefore, $15.\%$ of $135$ is

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