Algebra

Given that , we want to prove the following: First, calculate : Now, calculate and : Now, substitute these into the right-hand side of the equation: Simplifying the right-hand side: This is exactly the same as .
Final Answer:
Thus, we have proven that:

Step 1: Subtract from both sides.

Step 2: Subtract 4 from both sides.

Final Answer:
The solution to the inequality is .

Differentiating both sides with respect to : Rearranging terms:

Rewriting the equation: Use substitution , then differentiate and solve. The solution is: Thus:

Take the natural logarithm of both sides: Differentiating both sides: Simplify to find .

The dot product of two vectors is given by: where is the angle between the vectors. The absolute value of is always less than or equal to 1, i.e., . Therefore: