Show that there are two turning points on the curve y= x3+ x1. Determine the nature of each turning pointusing the first derivative method

Sure, let's solve this step by step.

Step 1: Find the derivative of the function The given function is y = x^3 + x. The first derivative of the function is obtained using the power rule (nx^(n-1)) for differentiation. So, the derivative of x^3 is 3x^2 and the derivative of x is 1. Therefore, the first derivative of the function y = x^3 + x is y' = 3x^2 + 1.

Step 2: Find the critical points The critical points of the function are the points where the derivative is zero or undefined. The derivative y' = 3x^2 + 1 is never undefined, so we set it equal to zero and solve for x to find the critical points.

0 = 3x^2 + 1 => 3x^2 = -1 => x^2 = -1/3

Since x^2 is always non-negative for real numbers, there are no real solutions to this equation. Therefore, there are no critical points, and hence no turning points for the function y = x^3 + x.

So, it seems there was a mistake in the question. The function y = x^3 + x does not have any turning points.