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4 years ago

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Which statement is true? A matrix separates rational and irrational numbers. A matrix separates positive and negative numbers. A

matrix orders numbers from greatest to least. A matrix is a rectangular arrangement of numbers.

1 answer:

Jet001 [13]4 years ago

5 0

I'm pretty sure the answer is "A matrix separates rational and irrational numbers." (Note: I'm not that great with memorizing definitions, but I'm pretty sure.) Hope this helps :)

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I have calculus problems that I need help with.

aleksklad [387]

a. Note that $f(x)=x^ne^{-2x}$ is continuous for all $x$ . If $f(x)$ attains a maximum at $x=3$ , then $f'(3) = 0$ . Compute the derivative of $f$ .

$f'(x) = nx^{n-1} e^{-2x} - 2x^n e^{-2x}$

Evaluate this at $x=3$ and solve for $n$ .

$n\cdot3^{n-1} e^{-6} - 2\cdot3^n e^{-6} = 0$

$n\cdot3^{n-1} = 2\cdot3^n$

$\dfrac n2 = \dfrac{3^n}{3^{n-1}}$

$\dfrac n2 = 3 \implies \boxed{n=6}$

To ensure that a maximum is reached for this value of $n$ , we need to check the sign of the second derivative at this critical point.

$f(x) = x^6 e^{-2x} \\\\ \implies f'(x) = 6x^5 e^{-2x} - 2x^6 e^{-2x} \\\\ \implies f''(x) = 30x^4 e^{-2x} - 24x^5 e^{-2x} + 4x^6 e^{-2x} \\\\ \implies f''(3) = -\dfrac{486}{e^6} < 0$

The second derivative at $x=3$ is negative, which indicate the function is concave downward, which in turn means that $f(3)$ is indeed a (local) maximum.

b. When $n=4$ , we have derivatives

$f(x) = x^4 e^{-2x} \\\\ \implies f'(x) = 4x^3 e^{-2x} - 2x^4 e^{-2x} \\\\ \implies f''(x) = 12x^2 e^{-2x} - 16x^3e^{-2x} + 4x^4e^{-2x}$

Inflection points can occur where the second derivative vanishes.

$12x^2 e^{-2x} - 16x^3 e^{-2x} + 4x^4 e^{-2x} = 0$

$12x^2 - 16x^3 + 4x^4 = 0$

$4x^2 (3 - 4x + x^2) = 0$

$4x^2 (x - 3) (x - 1) = 0$

Then we have three possible inflection points when $x=0$ , $x=1$ , or $x=3$ .

To decide which are actually inflection points, check the sign of $f''$ in each of the intervals $(-\infty,0)$ , $(0, 1)$ , $(1, 3)$ , and $(3,\infty)$ . It's enough to check the sign of any test value of $x$ from each interval.

$x\in(-\infty,0) \implies x = -1 \implies f''(-1) = 32e^2 > 0$

$x\in(0,1) \implies x = \dfrac12 \implies f''\left(\dfrac12\right) = \dfrac5{43} > 0$

$x\in(1,3) \implies x = 2 \implies f''(2) = -\dfrac{16}{e^4} < 0$

$x\in(3,\infty) \implies x = 4 \implies f''(4) = \dfrac{192}{e^8} > 0$

The sign of $f''$ changes to either side of $x=1$ and $x=3$ , but not $x=0$ . This means only $\boxed{x=1}$ and $\boxed{x=3}$ are inflection points.

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1 year ago

Read 2 more answers

Which expressions are equivalent to 6 +(-x) + 2x + (-7)

Fantom [35]

Answer:

a and c

Step-by-step explanation:

3 0

3 years ago

Identify the coefficient of each term of the polynomial.

fredd [130]

There are two parts of an equation (including this one), the coefficient, and the variable.
The coefficient is a known number, written as a digit, whereas the variable is an unknown number, written as a letter to signify an unknown number.

So, the coefficient in the first term (or part of the equation) is -2. The coefficient in the second term is 9, and the variable in this equation is x.

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3 years ago

What is the measure of the missing angle?

nekit [7.7K]

Answer:

80

Step-by-step explanation:

A triangle always measures to be 180 degrees. 50+50=100 so 80 degrees is left

5 0

3 years ago

Read 2 more answers

What's the estimated to 538,000? A( 553,255 B) 897,300 c) 537,334 d)600,000

skelet666 [1.2K]

Answer:

C. 537,334

Step-by-step explanation:

It is the closest estimation towards 538,000, since 537,334 rounded to the nearest thousandth would be 538,000.

Hope this helps!

Panna

4 0

3 years ago