Problem number 29 states, let m of k be the maximum value of the absolute value of 4x squared minus 4x plus k, for x in the closed interval from minus 1 to positive 1, where k can be any real number. What is the minimum possible value of m of k? So here's the function m of k that we were asked to figure out the minimum value of. The function takes the maximum of all the points on the closed interval from minus 1 to 1 of the absolute value of 4x squared minus 4x plus k. So in this problem, it's easy to get lost. So first, let's figure out how to find the maximum of the quadratic. So here, we're only considering the interval from minus 1 to positive 1. And here is some parabola on the interval. Now for any parabola on a closed interval, the maximum, or even minimum for that case, can appear only in three places. It can appear at either of the endpoints or at the vertex of the parabola. In this case, it appears at the vertex. But if the parabola is concave up, or if the interval doesn't include the vertex, well then the parabola's maximum would appear at the endpoints. Either way, with that settled, let us try to find where the parabola's vertex will lie on the x-axis. To do this, let's use a formula that the vertex's x-coordinate is equal to minus b over 2a. In this case, b is equal to negative 4, and a is equal to 4. So we get a fraction of 4 over 8, or 1 over 2. So we have a parabola whose function is defined by y equals 4x squared minus 4x plus k. Let's put it into vertex form. So let's isolate the x variables and ensure that the coefficient on x is going to be 1. Now let's add 1 fourth to both sides so we can factor the right side to minus 1 half. So after factoring, this is what we get. Multiplying both sides by 4. And now let's add k and subtract 1. So now we have three values of x to test. Minus 1, the lower bound, 1 half, the vertex, and 1, the upper bound. So let's sub each of our x values into the parabola. Simplify them down, and this is what we end up getting. The absolute value of k plus 8, k minus 1, and k. So the function m of k finds the maximum of these three values. And we need to find the minimum value of any k. So this is going to occur when the largest positive value is going to equal the lowest negative value. So in this case, let's set the absolute value of k plus 8 equal to the absolute value of k minus 1. And when we break the absolute value signs, let's say that the first one is going to be positive and the second one is going to be negative. So now we can solve for k. So we get that 2k is equal to negative 7. And solving for k, we get that k is equal to minus 7 halves. So the minimum value occurs when k is equal to minus 7 halves. Plugging that into our m equation, we get a value of 9 halves. So the question asks us, what is the minimum possible value of m of k?

minimum value

absolute value

parabola

vertex

interval