Various Forms Of The Equation Of A Line

We are given that and is maximum. To maximize , we use the method of Lagrange multipliers or simply apply the optimization conditions for a product. Let . Then, the function to be maximized is: To maximize this, we take the derivative of and set it to zero: Simplifying: Setting , we get: Substituting into , we get . Thus, and . Now, calculate : \bigskip

We are given that satisfies the conditions of Rolle’s Theorem. Rolle’s Theorem states that if a function is continuous on the closed interval , differentiable on the open interval , and , then there exists a point such that . Given: - - The roots of the equation are and . Step 1: Use the fact that and . Since , substitute in the equation : Similarly, substitute into : Step 2: Solve the system of equations. From Equation 1: From Equation 2: Multiply Equation 1 by 4 and Equation 2 by 5 to eliminate : Add Equation 3 and Equation 4: Substitute into Equation 1: Step 3: Find the value of . Now, differentiate : Substitute and : Since , solve for : Use the quadratic formula: Thus, one of the values of is . \bigskip