Use implicit differentiation to find 𝒅𝒚𝒅𝒙 if a) 5y2+siny= lnx2 b) 5y3-tany=ln2x-2x2In Problem (12-16), Finda) interval on which f is increasingb) interval on which f is decreasingc) open interval on which f is concave upd) open interval on which f is concave downe) the x-Coordinates of all infection pointsf) critical points

Use implicit differentiation to find 𝒅𝒚𝒅𝒙 if a) 5y2+siny= lnx2 b) 5y3-tany=ln2x-2x2

Consider the equation below. (If an answer does not exist, enter DNE.)f(x) = e9x + e−x(a)Find the interval on which f is increasing. (Enter your answer using interval notation.)(−ln(3)5​,∞) Find the interval on which f is decreasing. (Enter your answer using interval notation.)(−∞,−ln(3)5​) (b)Find the local minimum and maximum values of f.local minimum value −ln(9)10​ local maximum value DNE (c)Find the inflection point.(x, y) =  DNE Find the interval on which f is concave up. (Enter your answer using interval notation.)(−∞,∞) Find the interval on which f is concave down. (Enter your answer using interval notation.

The first derivative of the function f is defined by f'(x) = (x2 +1)sin(3x - 1) for -1.5 < x < 1.5. On which of the following intervals is the graph of f concave up?Responses(−1.5, −1.341) and (−0.240, 0.964)(−1.5, −1.341) and (−0.240, 0.964)(−1.341, −0.240) and (0.964, 1.5)(−1.341, −0.240) and (0.964, 1.5)(−0.714, 0.333) and (1.381, 1.5)(−0.714, 0.333) and (1.381, 1.5)(−1.5, −0.714) and (0.333, 1.381)

For each of the following two functions, complete all of the items below:y = −x3 + 3x2 + 144xy = 3x2 − 12x − 15(a) (3% ea) Find the x- and y-intercepts.(b) (3% ea) Find the derivative function.(c) (3% ea) Use the derivative to find any local maximums or local minimums.(d) (3% ea) Find the second derivative function.(e) (3% ea) Use the second derivative to find all inflection points.(f) (3% ea) Construct an x/y table using all points found above.(g) (2% ea) Use your table, a computer graphing tool or your graphing calculator, and the interpre-tation of your calculations to sketch an accurate graph of each function.2. (6% each part) Suppose we have opened a catfish hot dog stand at the mall which will be open 4 hoursper day. We are test marketing our sales at various prices to attempt to determine the best price weshould use for our catfish hot dogs. So, let:x = Number of y = Price of thehot dogs sold hot dogs186 $1.50128 $3.0051 $4.00Notice that we have put these variables in the same order that we did in Section 12, which is differentfrom what we have done earlier in the course.(a) Draw a scatterplot of these data values. Is it positively correlated, negatively correlated, or doesit exhibit no correlation? What is the correlation coefficient for this model?(b) What is the linear regression equation which fits this model? What is its slope? Give a verbalinterpretation of the slope.(c) Find the equation of the Revenue function R(x) for this model, and the equation of the MarginalRevenue function M R(x) for this function. Find the number of hot dogs sold that will maximizeRevenue. What would be your daily revenue at this number of hot dogs? What is the price youshould charge in order to generate this profit?(d) Using the following Costs:• $400 per month franchise fee1• $200 per month mall rental fee• $0.70 per hot dog from the distributor• $0.25 per roll from the distributor• $9.50 per hour to pay the employeeFind the equation of the Cost function C(x) for this application. You should assume that thereare 30 days for each month, and find the costs per day.What are the Fixed Costs? What are the Variable Costs?(e) Find the equation of the Profit function P (x), and the equation of the Marginal Profit functionM P (x).(f) Find the number of hot dogs you would have to sell per day to in order to maximize your profit.What will be your daily profit at this number of hot dogs? What is the price you should chargein order to generate this profit?(g) What is the Marginal Profit if you sell 160 catfish hot dogs per day? Interpret this number.(h) Is your profit increasing or decreasing when you sell 100 catfish hot dogs? Explain how youdetermined this.(i) Find all Break-Even points for this application.

he graph of a function f is given. Use the graph to estimate the following.(a) All the local maximum and minimum values of the function and the value of x at which each occurslocal maximum     (x, y) = 0,12 local minimum      (x, y) = −9,−6 (smaller x-value)local minimum (x, y) = 6,3 (larger x-value)(b) The intervals on which the function is increasing and on which the function is decreasing. (Enter your answers using interval notation.)increasing     (−∞,−9)∪(0,6) decreasing     (0,6)