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\{uestion 1 Given \[ f(x)=-x^{4}+2 x^{3}-3 x^{2}+7 x+9 \] find \( f(-1) \)....

Question

\{uestion 1 Given \[ f(x)=-x^{4}+2 x^{3}-3 x^{2}+7 x+9 \] find \( f(-1) \).



Answers

Find $f^{\prime}(x)$ $$ f(x)=x^{-3}+\frac{1}{x^{7}} $$

All right. So question 50 for Givin AF of X is equal to two X cubed minus seven X plus one. We want to find F of a half. All right, negative a half. Sorry. So, one thing we can do is you can use synthetic division good X cube zero X squared negative seven X. M one, whatever the remainder is from our division is the value of the function using what's called the remainder theorem. So that's negative one. That's negative one. That's positive one half. This is -13 over to negative half is 13/4 plus four. Four is 17/4. So F of negative a half equals 17/4.

Okay, so we're starting with the formula F of X is equal to four X plus 9/2. We're looking for its inverse A We're going to switch the X and the y So first recesses Why equals four X plus nine divided by two more. Switch the X and Y locations. And now we're gonna solve for solve for y So we're in a multiply both sides by two We're gonna subtract nine from both sides and then we're gonna divide everything by four. So the inverse that the notation will be two X minus 9/4.

Okay, so we're asked to find the fallen. So we're gonna need are derivative for our function. And we also need to find f in verse. That's a well, our A is equal to four. That's the same as a of X Equalling. For so that is two X cubed plus three X squared. What's seven X and then plus four is equal to four. Okay, so what X value gives us for what we know that if X is zero gets plus for so f of geral is equal to four. So we have that are in verse that's four is equal to zero. Okay, so now that we've drawn that, let's find our derivative for function of prime of X that you could to six or three times to, which is six x squared plus, um six X plus seven. OK, And now let's evaluate that at, um, at this point here, actually, about that one, um, at this point here tonight. So So we get, um, zero plus several plus seven, that is, our solution is equal. Teoh 1/7

Okay, So this form wants you to find the derivative of the function exterminated three plus one over Exodus Seventh. So Well, what we will essentially be doing here. But this problem is using the power role now first, what we want to do to make things easier and let look nicer is to rewrite this equation as extra naked three plus extra the negative seven. Because one over X and seventh equals two extra negative seven. Since the negative excellent exponent equals to the reciprocal. And this makes it easier because the power rule the power will if we were to take the derivative of power from two using the power. It's not that simple. Using the power rule, we just have to bring down the exponent and and then multiply by X to the n minus. What so we have to do now is applied this rule to both of these terms X in 83 1987. So first we have extra negative three to bring down the negative three, and we'll deploy it by X to the negative four since made a B minus wise native for and we have to summit cause the derivative exit Negative. Seven. Well, we have toe using the power rule. We have to bring down the negative seven and multiply by X to the negative eight. Since negative seven miles, one is negative. Eight. So your function your F prime of X, you close to negative three x a negative for minus seven times X two, native eight and this is your final answer.


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