Find the maximum and minimum values of the function f(x) = x^3 - 7x^2 + 8x - 1 on the interval [0, 6].
We'll begin by using the first derivative of the function to find any local maxima and minima in the interval. The critical points where maxima and minima may exist are found either where the function is discontinuous or where the first derivative is equal to 0. The function is a polynomial, so it is not discontinuous anywhere over the real numbers. This leaves us with critical numbers found from the first derivative: f'(x) = 3x^2 - 14x + 8 0 = 3x^2 - 14x + 8 0 = (3x - 2)(x - 4) Critical points are found where x = 2/3 and where x = 4 By finding the values of the functions at these points, as well as at the endpoints of the interval, we can find the maximum and minimum values in this interval: f(2/3) = 41/27 or approximately 1.52 f(4) = -17 f(0) = -1 f(6) = 11 The minimum value is -17, found at one of the critical points, while the maximum value is 11, found at the endpoint where x = 6.
Choose the option that best replaces the bracketed portion of the sentence. Both City Hall and Holtzmann Tower, the tallest building downtown, [was rebuilt] following the earthquake. A) no change B) were rebuilt C) rebuilt D) had been rebuilt
The correct choice is B, The subject is compound since it includes both City Hall and the Holtmann Tower. The verb must agree with the subject, and therefore should be plural. The verbs for choices A and D are both singular. The verb for choice C is active, which would imply the buildings did the rebuilding themselves.
Find the x- and y-intercepts of the following line: 3x - 2y = 8
A line has x- and y-intercepts where it crosses the x- and y-axes, respectively. Any point on the x-axis will always have a y coordinate of 0, so we can find the x-intercept by substituting 0 for y in the line: 3x - 2y = 8 --> 3x - 2(0) = 8 3x = 8 x = 8/3 So the x-intercept is (8/3, 0). The same process can be applied to finding the y-intercept, this time substituting 0 for x: 3x - 2y = 8 --> 3(0) - 2y = 8 -2y = 8 y = -4 y-intercept: (0, -4)