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- A function y = f(x) is decreasing on the interval [0, 5]. What conclusions can you draw about the graph of y = f -1 (x)?find the critical points and and absolute extrema of the function y=9-x^2 on the interval [-3,1]The U.S. Bureau of the Census prediction for the percentage of the population 65 years and older can be modeled as p(x) = −0.00022x3 + 0.014x2 − 0.0033x + 12.236 percent where x is the number of years since 2000, data from 0 ≤ x ≤ 50. 1. Determine the value of x in the domain 0 ≤ x ≤ 50 for which the percentage is predicted to be increasing most rapidly (Round your answer to three decimal places). 2. Thus, in which year is the percentage is predicted to be increasing most rapidly? 3. Calculate the percentage at that time. (Round your answer to two decimal places.) 4. Calculate the rate of change of the percentage at that time. (Round your answer to three decimal places.)
- tent line for h(x)=(2x^(2)+8x-5)^(2) at the x-value x=-1.A hat manufacturer is planning to expand its workforce. It estimates thatthe number of hats produced by hiring x new workers is given byT(x) = – 0.25 x4 + 6 x3 0 ≤ x ≤ 18a) When is the rate of change of hat production increasing and when is itdecreasing?b) What is the point of diminishing returns (Hint: inflection point) and themaximum rate of change of hat production?The oxygen supply, S, in the blood depends on the hematocrit, H, the percentage of red blood cells in the blood. If S = k(H) = aHe-bH for positive constants a and b, with domain (0, infinity) and k'(H) = ae-bH(1-bH) 1. Use the definition to find the only critical point H1 of k on its domain. 2. Use a number line and the first derivative test to show that the oxygen supply is maximised at H1.You have to explain how you determine the sign of the first derivative on every interval. 3. What is the maximum oxygen supply? 4. How does increasing the value of the constants a and b in the same proportion change the maximumvalue of S? Please answer 3 and 4
- 2.Explain why f(x) = x/(x-2) does not have an extreme value on the interval (0,5).4.The derivative of a function f is given by f'(x)=(-2x-2)e^x, and f(0) = 3.A. The function f has a critical point at x = -1. At this point, does f have a relative minimum, a relative maximum, or neither? Justify your answer.B. On what intervals, if any, is the graph of f both increasing and concave down? Explain your reasoning.C. Find the value of f(-1).Q2:Find all critical points of the function f(x)=(3x)/[(x^2)+9] Final answers should be ordered pairs.
- 3- When calculating the approximate solution of the f(x)=0 equation in the interval [a,b] using the graphical method, which of the following is accepted as the solution?a)The point of the function graph that is closest to the axis. B) The point where the graph of the function cuts the y-axis.NS)The point of the function graph closest to the x-axis. D)The point of the function graph that is equidistant from the x and y axes. TO)The point where the graph of the function intersects the x-axis.A problem that occurs with certain types of mining is that some byproducts tend to be mildlyradioactive, and these products sometimes get into our freshwater supply. The EPA has issuedregulations concerning a limit on the amount of radioactivity in supplies of drinking water.Particularly, the maximum level for naturally occurring radiation is 5 picocuries per liter ofwater (on average). A random sample of 24 water specimens from a city’s water supply gave ̄x= 4.61 and s = 0.87. Do these data provide sufficient evidence to indicate that the meanlevel of radiation is safe (below the maximum level set by the EPA)? Test usingα= 0.05.Use algebraic methods to determine the critical value(s) of f(x)= x/(x^2-x-2). Give your answers in exact form.