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Find the value of € that makes the function continuous on R ~cx2 + 2 if c < 5 f(c) = 3c _ 322 if c > 5Answer:The smaller value of € isThe larger v...

Question

Find the value of € that makes the function continuous on R ~cx2 + 2 if c < 5 f(c) = 3c _ 322 if c > 5Answer:The smaller value of € isThe larger value of c is

Find the value of € that makes the function continuous on R ~cx2 + 2 if c < 5 f(c) = 3c _ 322 if c > 5 Answer: The smaller value of € is The larger value of c is



Answers

Find the value of the constant $(a, b,$ or $c)$ that makes the function continuous. $$ f(x)=\left\{\begin{array}{ll} x^{2}-c & \text { for } x<5 \\ 4 x+2 c & \text { for } x \geq 5 \end{array}\right. $$

The question we have you been the f acts ik 02 x squared minus C for the X monitor and five and far ex pressed, juicy excreted equal to five And we need to find this cease A standard function is continues are everywhere So in order to do that, we need to make so the limit on the function when ex prostitute five from the left must encourage Italy mitt on a function experts to under five from the right, I must take urgent effort Five we see an effort five we had the municipal isn't the function here and then we should get echo to uh ah for times five plus to see an ecology gente breast to see And now the limit on the function When x goes Teoh five from the left and we're go generally mitt off Ah, from the left. And then how do you displace here? So we have the extreme honest see extra Jew fire from the left And then we could you want to find your this one and again it into five minus e and a limit on the function when executed, choosing from the right exactly equal to this F five. So we want to make so that least you here must be equal. So with equal now and then what? You need to serve this question here on and hey, I was getting Now it will be We bring the seat here to here, to the right and 20 head to the left. They were gonna five year Coggiola, three c. Therefore, we have to say we could your five out of three and that will be in a value of just under functional. Continuing continues everywhere.

For the given problem we want to find the minimum cost C. N. Dollars. Given that five times C minus 25 is going to be greater than or equal to 1.75. Mhm. Class 2.5 C. Yeah. So we're going to solve for C. Here um Knowing that this makes five C. And this makes negative 1 25. So we're going to add 125 to both sides And that makes this 126 .75. And then we subtract this, making this 2.5 c. And then lastly we'll divide both sides by 2.5 126.75, divided by 2.5 is 50.7. So we see that C. is greater than or equal to 50.7 and that's gonna be our final answer. That's gonna be the minimum cost.

Our goal for this problem is to solve the inequality and find minimum cost. So we're given that five times c minus 25 is less than or equal to 1.75 plus 2.5 seats. So as we've discussed before, let's treat this is somewhat of an equal sign. So we're going to get all the terms on one side. All terms on another side combined are like terms divide basically isolate see in the same way we would do when solving an equation. So let's start by multiplying this through. We'll get five c minus 1 25. Then we can add the 1 25 over here, making this 1 20 6.75 and then we'll subtract the negative, will subtract 2.5, making this 2.5 C. We know that 1 26.75 divided by 2.5 is going to be 50.7 so we'll end up getting that. See the cost must be greater than 50.7. Final answer.

Okay in this problem we have f of x, Y piecewise defined to be C plus Y. We're next less than or equal to three. That's going to be to the left of the X equals three line. And our function is defined to be a five minus fly when x is greater than three when we are on the right side of the red x three line. So on the left side of the red line uh F equals C plus y on the right side of the red line. Uh to function equals five minus white. Now we want to find out if this function is if we can find the value of C, we want to find out if we can find the value of C. Uh a constant number for say, that would make this function continuous everywhere on the xy plane. Well, if we approach uh this point at the end of this blue arrow along this blue line, the y coordinate is staying fixed at three. So the white coordinators tree on this blue line. Now we're on the left side of this red line, so X less than or equal to three. So our function along this blue line equals C. Plus. Why? Now along this blue line our function equals C. Plus why? Uh But why is fixed at three? So our function F equals C plus straight everywhere on this line. So F is approaching the value of C plus three as we approach this point at the tip of the arrow, Okay, uh left side of the red line, X less than or equal to three. Our functions to find A B. C plus Y. The blue line is in this part of the region. So our function on the blue line equal C plus Y, C. Plus the Y coordinate. Now on this blue line, the y coordinate, a very point on this blue line is straight, so F equals C plus y, F equals C plus three, everywhere on this blue line. So as we approach this point at the tip of the arrow uh is approaching the value of C plus three, mainly because of equal C plus three everywhere on that blue line. Now if we come in uh approaching the same point, but from the right direction now in on in this region, okay to the right side of the red line, X is greater than three. Uh So for points on this black line are functioning equals five minus y. No so everywhere on this black line our function equals five minus white. But everywhere on this black line um the y coordinate is three again, so F equals five minus y or five minus three. Which is to so F is approaching five minus three, which of course equals two. Um as we approach this point um from the right direction, as we move along this black line to the tip of the black arrow at this point right here, at the tips of the blue and black arrows, uh F is equal to five minus y. Um every point on this black line, why is three, so F is equal to five minus three, which is to everywhere on this black line. So since f is staying fixed at two on this black line, F of course approaches the value of two as we move to this point here at the end of the black line. So in order for F to be continuous at this point, uh F would have to be approaching the same value as it approaches this same point from the two different directions, so C plus three would have the equal to, So we need C plus three, two equal to for our function F of X. Y to be continuous. At this point, at the end of these activities Arabs we would need C. Plus to read equal to. So let's keep that in mind. No. What if we approach uh the point at the tip of this arrow where every point along this green line? Uh the y coordinate is staying fixed at one? Well, since we are in the region where X is less than or equal to three to the left of the red line, F equals C. Plus why? Uh C plus Y is going to be C plus one along this red line because the y coordinate stays one. So C plus Y is C plus one everywhere on this green line. So since F stays fixed at C plus one on this green line, as we approach this point at the tip of the green arrow of his, approaching the value of C plus one. No uh Let's just use a different colour. Now as we approach the point at the tip of that green arrow, which is now the point at the tip of the green and red arrows, we are in the region where X is greater than three. Uh So our function on this red line is equal to five minus y, so F is equal to five minus y. But of course on this red line, the y coordinate is fixed at one, so F equals five minus Y. F equals five minus one, which is four on this red line. So F is equal to four everywhere on this red line. So F approaches the value of four since it stays equal to four. Um as we move along this red line to the point at the tip of the red arrow. So for the function F to be continuous at the point here, which is at the tip of the green and red arrows, uh F would have to be approaching the same value as we approach the same point from the two different directions, so we would need C plus one, which is what F approaches from this direction to equal for which is the value that F approaches from this direction. So we need uh C plus one to equal four, so we need both of these equations to be satisfied. Well, if C plus three has to equal to, that means C has to equal negative one, but uh it's C plus one equals four. That means C equal strength. Well, in order for the function to be continuous everywhere on the xy plane, in particular for the function uh to be continuous at these two points, we need to see to be equal to negative one, but it also has to equal three. C is a constant. It can't equal to different numbers, so there is no value of C that will make this function uh continuous everywhere on the xy plane.


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