<img src="https://tex.z-dn.net/?f=%20-%20%20%5Cfrac%7B17%7D%7B6%7D%20x%20%2B%201%20%3D%20%20%5Cfrac%7B7%7D%7B6%7D%20%20-%20%20%5

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3 years ago

Cfrac%7B5%7D%7B2%7D%20x" id="TexFormula1" title=" - \frac{17}{6} x + 1 = \frac{7}{6} - \frac{5}{2} x" alt=" - \frac{17}{6} x + 1 = \frac{7}{6} - \frac{5}{2} x" align="absmiddle" class="latex-formula">
can someone help me solve this?
and also explain how you got the answer too
thank you

Mathematics

1 answer:

Shalnov [3]3 years ago

3 0

$-\dfrac{17}{6}x+1=\dfrac{7}{6}-\dfrac{5}{2}x\\\\\text{Most people do not like fractions. Well, let's get rid of them.}\\\\\text{Find the common denominator.... It's 6 :)}.\\\\\text{Multiply both sides of the equation by 6:}\\\\6\!\!\!\!\diagup^1\cdot\left(-\dfrac{17}{6\!\!\!\!\diagup_1}x\right)+6\cdot1=6\!\!\!\!\diagup^1\cdot\dfrac{7}{6\!\!\!\!\diagup_1}-6\!\!\!\!\diagup^3\cdot\dfrac{5}{2\!\!\!\!\diagup_1}x\\\\-17x+6=7-3\cdot5x\qquad\text{subtract 6 from both sides}$

$-17x=1-15x\qquad\text{add 15x to both sides}\\\\-2x=1\qquad\text{divide both sides by (-2)}\\\\\boxed{x=-\dfrac{1}{2}}$

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The fifth degree Taylor polynomial of g(x) is increasing around x=-1

Step-by-step explanation:

Yes, you can do the derivative of the fifth degree Taylor polynomial, but notice that its derivative evaluated at x =-1 will give zero for all its terms except for the one of first order, so the calculation becomes simple:

$P_5(x)=g(-1)+g'(-1)\,(x+1)+g"(-1)\, \frac{(x+1)^2}{2!} +g^{(3)}(-1)\, \frac{(x+1)^3}{3!} + g^{(4)}(-1)\, \frac{(x+1)^4}{4!} +g^{(5)}(-1)\, \frac{(x+1)^5}{5!}$

and when you do its derivative:

1) the constant term renders zero,

2) the following term (term of order 1, the linear term) renders: $g'(-1)\,(1)$ since the derivative of (x+1) is one,

3) all other terms will keep at least one factor (x+1) in their derivative, and this evaluated at x = -1 will render zero

Therefore, the only term that would give you something different from zero once evaluated at x = -1 is the derivative of that linear term. and that only non-zero term is: $g'(-1)= 7$ as per the information given. Therefore, the function has derivative larger than zero, then it is increasing in the vicinity of x = -1