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Problem 80 Hard Difficulty

Assume that all of the functions are twice differentiable and the second derivatives are never $ 0 $.

(a) If $ f $ and $ g $ are concave upward on $ l $, show that $ f + g $ is concave upward on $ l $.
(b) If $ f $ is positive and concave upward on $ l $, show that the function $ g(x) = [f(x)]^2 $ is concave upward on $ l $.

Answer

a) While differentiating $f+g,$ we used $(f+g)^{\prime}=f^{\prime}+g^{\prime}$
Therefore $f+g$ is concave upward on $I$
b) $g^{\prime \prime}(x)=2 f(x) \times f^{\prime \prime}(x)+2\left[f^{\prime}(x)\right]^{2}$
since $f(x)$ and $f^{\prime \prime}(x)$ are positive, $g^{\prime \prime}(x)$ is positive too.
Which means $g(x)$ is concave upward

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Fazal S.

February 7, 2021

Assume that both the functions f and g are twice differentiable and the second derivatives are never 0 on an interval I. If f and g are concave upward on I, show that f + g is concave upward on I

Video Transcript

So we're told that F and G are both twice differentiable, and also that their concave up on this interval. So on interval I and what we want to find out, what we want to show, I guess is that F plus G is also concave up on this interval. And so the way that we're gonna do that is we know that F plus G is just the summation of these two functions. So the derivative of F plus G. Well, when we find when we find the derivative of some function plus another function, that's just equal to the derivative of the first, plus the derivative of the second. So this is just equal to F prime plus G prime. And then if we were to do this again, let's just say F prime plus G prime. And then we found these next derivative which would be our second derivative. This will just be equal to F double prime plus G double prime. And since we know that F double prime and G double prime are both going to be positive. Since we're concave up on this interval, that means that F plus G the second derivative of F plus G is also going to be positive, which means that we're going to be concave up for the function F plus G as well. So, since our second derivative of F plus G is greater than zero, F plus G is concave and for part B, what we're gonna do is gonna show that if some function F is positive and concave up. So we're gonna say F is positive and I'm going to denote concave up as see you on the interval. I then we need to show that this function G of X, which is just equal to or function F of X squared. We need to show that that is also concave up on the interval. Yeah, I or l It might be L actually, but so if we go, go ahead and do this, what we're gonna need to do is find the first derivative of G of X. And so we're just going to use the chain rule to find this. So we need to bring down this to first. So he brings around that, bring down the two. You have two times F of X, two, now two times one or two minus one, which is to the first. And then we have to multiply by the derivative of F of X. So we have to use the chain role here and multiply by the derivative of fx after we bring down this exponents. And so now that we found this first derivative we're going to do is find the second derivative, which is going to be using the product rule. So we're gonna use we're gonna do the derivative of the first. So we have two times if prime of X times the second, which is also f prime of X. So this is F prime of X squared plus the first, which is just F of X times the derivative of the second, which would be F prime or F double prime of X. So if we simplify this is equal to two times F prime of X squared plus F of X times F prime or F double prime of X. And then this is also multiplied by two. I forgot the two. So if we look at this second derivative of G of X, this term here is two which is positive times some number or F prime of X is just gonna be some number squared in any number squared is going to be positive. So this is going to be a positive number. Two times F prime of X squared. This is gonna be a positive number since we don't have any eyes or imaginary numbers in our domain. And now, if we look at the second derivative or the second term in our second derivative, we know f of X is positive, where you're told f of X is positive on this interval. And we also know that f of X is concave up on this interval, so we know that it's second derivative is also going to be positive. So this term is going to be two times a positive number since f of X is positive and then times another positive number since we're concave up. So this is going to be equal to a positive number plus a positive number, which means that G prime or G double prime of X is going to be greater than zero on the interval I, which means that G G double prime of X is concave up