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GnuerstanaingIn Problems 74-76, explain what is wrong with the statement; 74, Ifw (r) = In (1+2' then W (r) = 1/ (1+2'). 75. The derivative of f(z) In (In...

Question

GnuerstanaingIn Problems 74-76, explain what is wrong with the statement; 74, Ifw (r) = In (1+2' then W (r) = 1/ (1+2'). 75. The derivative of f(z) In (Inx) isf' (c) ZInx + Inz? 21nf_ 76 _ Given f(2) = 6, f' (2) = 3,and f-1 (3) 1 we have(f ') (2) Ti("()) 4 In Problems 77-80. give an example of: 77: A function that is equal to constant multipl of its derivative but that is not equal to its derivative: 78. funetion whose derivative is c/x;where conslant. 79. function

Gnuerstanaing In Problems 74-76, explain what is wrong with the statement; 74, Ifw (r) = In (1+2' then W (r) = 1/ (1+2'). 75. The derivative of f(z) In (Inx) is f' (c) ZInx + Inz? 21nf_ 76 _ Given f(2) = 6, f' (2) = 3,and f-1 (3) 1 we have (f ') (2) Ti("()) 4 In Problems 77-80. give an example of: 77: A function that is equal to constant multipl of its derivative but that is not equal to its derivative: 78. funetion whose derivative is c/x;where conslant. 79. function Jx) for which f' (c) f' (cr) where cunstant. 80. function f sueh that 40-' f() Are the statements in Problems 81- Inic or false? Give an explanation for your MsTer 81. The graph of In (1?) is concave Up for



Answers

In Problems $74-76,$ explain what is wrong with the statement. For any function, $\int_{1}^{3} f(x) d x$ is the area between the the graph of $f$ and the $x$ -axis on $1 \leq x \leq 3$.

So first we want to consider F and G are positive, increasing in concave upward. So we want to show that their function is concave upward. So the function, is it going to be F prime of X. Give X. Plus? Um Uh huh. I am uh proposed aftereffects G prime of X. We need to take the second derivative. So they're not going to be F double prime of X. G F X. Class F effects the plus at the crime of X. Do you prima backs? So with that that's gonna be the first term derivative and then we'll do something very similar for the second term and will apply all the knowledge that we know about the fact that it's increasing its con cavity to find that it is. In fact, we can't give up word greater than zero.

That's incorrect because they multiplied F and G together to get the composite, but that's not how we do composite. So first you want to take the function F. And substitute G for every X value. Can we have systems are excess. First we substitute X squared minus four in tex. We had to don't you just simplify will be X squared minus two. Then we take the domain. We see that there's no denominator. No, no radical. So our function is going to go negative in the to infinity since there's no restriction.

This exercise. We're dealing with the properties of functions with her twice differential and have second derivatives that air never zero now in part, a rest to show that if two functions f N g or con cave upward on interval I then it follows that f plus g is concave upward on the interval I to prove this statement we'll use the converse of the Comte cavity test So we have it if f n g our con cave upward on the interval, I yeah, Then it follows by the converse of the concave ity test. That f double prime is greater than zero and gettable prime is greater than zero on the interval I Therefore, it follows that f plus g double prime which is equal to s double prime plus G double prime is greater than zero plus zero which is zero on I and therefore by the con cavity test. Yeah, we also know that F plus G is Khan cave upward on the interval, I in part me basket prove that if f is a positive and Kong cave upward on the interval I than the function G of X, which is the square of F is Khan Cave upward. Also on I. So yes, F is positive and Khan Cave upward on I and it follows that by definition, F is greater than zero and by the converse to the common cavity test F double prime is greater than zero on I. So it follows that G prime, which by the chain rule is equal to two times f times F prime and G double prime. The second derivative by the product rules is two times f times f double prime plus two times f prime squared. And we see that since at this positive and ethical prime is greater than zero and f prime squared is automatically greater than or equal to zero. This expression is going to be greater than zero on I and therefore by the con cavity test it follows that G itself, which is f squared. Is Khan cave upward on the interval? I

It's probably wanna explain what is wrong with this statement. The function that is not concave down, Yeah. Is concave yeah. Concave down means that F and double prime is less than zero. I gave up. That means that if double prime is greater than zero, and so just from the surface here, it would seem to appear that one of these always has to be true. But the problem is, it doesn't take into consideration when F double crime is zero and so this is false because it does not consider one. f double prime is zero. Yeah, If f double prime is zero mm than F is neither. Mhm khan gave up nor concave down.


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