Question

Find the Local Maxima and Minima f(x)=x²+2x-3

a) Maxima at x=−1, Minima at x=−2
b) Maxima at x=−2, Minima at x=−1
c) Maxima at x=1, Minima at x=−2
d) Maxima at x=−1, Minima at x=2

Answer 1

To determine the function's local minima, we differentiate the function, set the derivative to zero, and apply the second derivative test. The function has a local minimum at x = -1 since the second derivative is positive, indicating a concave up parabola. There is no local maximum within the real number domain.

Step-by-step explanation:

To find the local maxima and minima of the function f(x) = x² + 2x - 3, we use calculus. First, find the derivative of the function, f'(x), then set the derivative equal to zero to find critical points. These critical points could be potential candidates for local maxima and minima.

The derivative of the function is f'(x) = 2x + 2. Setting this equal to zero gives us x = -1 as a critical point. We use the second derivative test to determine the nature of this critical point. The second derivative is f''(x) = 2, which is always positive, meaning that the graph is concave up and the critical point at x = -1 is a local minimum. Since this is a parabola, and we've found a minimum with a positive second derivative, there can be no local maximum for this function within the real number domain.