1. Energy consumption.The typical daily energy use of a household is described by a function ƒ : [-12, 12] → [0, ∞).The value of f(x) is the consumption rate at given time x. Both points x = −12 and x = 12of the domain correspond to 3am so ƒ(12) = f(-12). It is also known that 9am and 9pm(x = −6 and x = 6, respectively) are local maxima of the consumption rate. In this questionwe model the rate by a polynomial of degree 4 that is ƒ(x) = axª + A3x³ + A2x² + a₁x + ªðand assume that the total energy consumption during the day equals 24.(a) Write a linear system of equations describing the following properties: ƒ(12) = f(−12),x = −6, 6 are stationary points of f and the total energy consumption is 24.(b) Using Gaussian elimination, find all solutions of the linear system obtained in (a).(c) Find all functions f that satisfy all of the data given in the description of the question.(d) Is it possible that the total energy consumed during the night hours from 1am to 5amis smaller than 2 (that is the average over these four hours is less than half of the dailyaverage)?If your answer is “yes” then you should give an example of such function f from (c),otherwise justify why it is not possible.

Since you have posted a question with multiple sub parts, we will solve the first three sub parts…

The typical daily energy use of a household is described by a function f [-12, 12] → [0, ∞). The value of f(x) is the consumption rate at given time x. Both points x = -12 and x = 12 of the domain correspond to 3am so f(12) = f(-12). It is also known that 9am and 9pm (x = -6 and x = 6, respectively) are local maxima of the consumption rate. In this question we model the rate by a polynomial of degree 4 that is f(x) = a4x4 + A3x³ + a₂x² + α₁x + ao and assume that the total energy consumption during the day equals 24. (a) Write a linear system of equations describing the following properties: f(12) = f(-12), x = -6,6 are stationary points of f and the total energy consumption is 24. (b) Using Gaussian elimination, find all solutions of the linear system obtained in (a). (c) Find all functions f that satisfy all of the data given in the description of the question.