Which equation has no solution? A.
24x + 32 − 2.1x = 20.9x + 30 + x + 2
B.
24x + 32 − 2.1x = 20.9x + 30 + x − 2
C.
24x + 32 − 2x = 20.9x + 30 + x − 2
D.
24x + 32 = 20.9x + 30

Solved

Show answers

Ответ:

slavenkaitlynn

27.05.2022

Mathematics

Substitute

for

to find that this function is equivalent to:

y^3 + ky^2 - 9y - 27

We take the derivative with respect to

to find the local minima, which occur at the solutions to:

3y^2 + 2ky - 9 = 0

Using the quadratic formula gives:

$y = \frac{-2k \pm \sqrt{4k^2 + 108}}{6}$

By the Trivial Inequality, since y=x^2

, $y \geq 0$ , so $y=\frac{-2k + \sqrt{4k^2 + 108}}{6} = \frac{-k + \sqrt{k^2 + 27}}{3}$ .

To determine whether this is a local minimum or a local maximum, we compute the second derivative, which is 6y+2k

. This is positive for all positive values of

(a reasonable assumption, since things get really messy really quickly otherwise), so the function is concave up at these values. Hence, this value is a local minimum.

Since it is the only local minimum, we conclude that it is the global minimum, since as

goes to positive or negative infinity, the x^6

term dominates and makes the function become extremely large.

The arithmetic gets pretty messy from here (I used WolframAlpha to ensure that I didn't make any mistakes), but in short, the rest of the problem pretty much devolves to substituting this value for

back in, which gives you $f(x) = \frac{2k^3}{27} + 3k - \frac{2}{27} \sqrt{(k^2 + 27)^3} - 27$ as the minimum.

Now, to show that there is no maximum, note that $\lim_{x \to \infty} f(x) = \lim_{x \to \infty} x^6 = \infty$ . Hence, the range of f(x)

is $[\frac{2k^3}{27} + 3k - \frac{2}{27} \sqrt{(k^2 + 27)^3} - 27, \infty)$ .

0,0(0 оценок)

Ask your question to the AI advisor

Ai-bot is an expert in any field and is the perfect companion for reliable and useful answers and advice on a variety of topics, including science, history, technology, art, sports, health, culture and more.

Ask your question

More tips

Answers on questions: Mathematics

Ask an AI advisor a question

I want advice

Which equation has no solution? A. 24x + 32 − 2.1x = 20.9x + 30 + x + 2 B. 24x + 32 − 2.1x = 20.9x + 30 + x − 2 C. 24x + 32 − 2x = 20.9x + 30 + x − 2 D. 24x + 32 = 20.9x + 30