Next, we find the critical points by setting \( R'(t) = 0 \):

["Title: Mastering Critical Points: Finding Critical Values by Solving ( R'(t) = 0 )", "In calculus and optimization, identifying critical points is a fundamental step in understanding how functions behave. One of the most powerful techniques for locating these points—and unlocking deeper insights into the function’s graph—is setting the first derivative equal to zero: ( R'(t) = 0 ).", "What Are Critical Points?\nCritical points occur at values of ( t ) where the function’s derivative is either zero or undefined. These points are critical because they often represent local maxima, local minima, or points of inflection—key features every dynamic model or cost-optimization problem seeks to identify.", "Why Set ( R'(t) = 0 )?\nThe derivative ( R'(t) ) measures the rate of change of the function ( R(t) ) at any point. When ( R'(t) = 0 ), the slope of the function is horizontal. This indicates a potential peak, valley, or saddle point—a behavior difference central to maxima and minima.", "Step-by-Step: How to Identify Critical Points by Solving ( R'(t) = 0 )", "1. Compute the Derivative\n Start by finding the derivative ( R'(t) ) of the original function ( R(t) ). This step requires standard differentiation rules (power rule, product rule, quotient rule, etc.), depending on the function's form.", "2. Set the Derivative Equal to Zero\n Solve the equation ( R'(t) = 0 ). This pedagogical cornerstone simplifies the infinite many points in the domain to just the critical candidates.", "3. Solve for ( t )\n Use algebraic techniques such as factoring, the quadratic formula, substitution, or more advanced solving methods to isolate ( t ).", "4. Verify the Points Exist in the Domain\n Ensure the solutions are within the domain of the original function ( R(t) )—this avoids extraneous or invalid solutions.", "5. Analyze the Critical Points\n Apply the first or second derivative test to classify each critical point:\n - First Derivative Test compares function values before and after ( t ), revealing whether a maximum, minimum, or neither occurs.\n - Second Derivative Test examines concavity at the point by evaluating ( R''(t) ), offering insight with minimal computation.", "Real-World Applications\nFrom maximizing profit functions to minimizing cost curves in engineering and economics, finding critical points by solving ( R'(t) = 0 ) enables precise optimization. It underpins key decisions in business strategy, physics modeling, and data science.", "Summary\nSetting ( R'(t) = 0 ) is a pivotal technique for uncovering critical moments in a function’s behavior. This approach forms the backbone of calculus-based optimization and is essential for anyone analyzing change, trends, or efficiency in mathematical and real-world systems.", "Remember: Don’t stop at solving ( R'(t) = 0 )—always test for extrema and ensure solutions lie within the function’s domain.", "---", "By mastering this method, you empower yourself to analyze functions deeply, optimize performance, and gain a strategic edge in calculus-driven problem solving. Keep practicing—each derivative solved brings you closer to maîtrise in continuous mathematics."]