Solve any maths problem

Step by step

1. f(x) = \sin{\left(\pi x \right)}

   Critical points are where f′(x) = 0 or is undefined.

2. f'(x) = \pi \cos{\left(\pi x \right)}

   Differentiate.

3. x = \frac{1}{2}, x = \frac{3}{2}

   Solve f′(x) = 0.

4. f''(x) = - \pi^{2} \sin{\left(\pi x \right)}

   Second-derivative test: f″ < 0 → maximum, f″ > 0 → minimum.

5. f''(\frac{1}{2}) = - \pi^{2} \Rightarrow (\frac{1}{2}, 1) \text{ is a local maximum}

6. f''(\frac{3}{2}) = \pi^{2} \Rightarrow (\frac{3}{2}, -1) \text{ is a local minimum}