Final answer:
To analyze the function y=31x3-x2+x+3 for relative maxima, minima, and points of inflection, calculate the first and second derivatives, finding critical points where the first derivative equals zero and identifying points of inflection by changes in sign of the second derivative.
Step-by-step explanation:
To find the relative maxima, relative minima, and horizontal points of inflection for the function y=31x3−x2+x+3, you need to first take the derivative of the function to find critical points. The critical points are where the first derivative is zero or undefined. The relative maxima and minima can be determined by analyzing the first and second derivatives. Points of inflection are found where the second derivative changes sign.
Step 1: Find the first derivative y' (dy/dx) which gives us the slope of the tangent at any point on the curve. Set y' to zero to find critical points.
Step 2: Find the second derivative y'' (d2y/dx2) which gives us the curvature. If y'' is positive, the original function is concave up, and if negative, it's concave down.
Step 3: Use the second derivative test to classify the critical points found in Step 1 as relative maxima or minima. Analyze the change in sign of y'' to identify points of inflection.