Finding the min and max values of a function and marking them on a graph using findmaximum/minimum

Question

This is what i have so far to find the points that are zero and mark them on the graph. But i cant for the life of me figure out how to proceed. Only started using mathematica last week. I need to mark the min and max values of the function too. But i dont know how to find them.

enter f[x_] = x * Cos[π/6]*(x^5 - 2 x^3 + x) - 0.1;

solution = Solve [f[x] == 0];

zero = Table [{x, 0} /. solution];

Plot [Ekv[x], {x, -1.2, 1.2}, PlotRange -> {-0.12, 0.05}, 
Epilog -> {Point /@ zero}]

Answer 1

The commands FindMinimum[] and FindMaximum[] are used to find local minima and maxima, respectively. In the current case, you can use Solve to get the stationary points directly.

 stationaryPoints = Transpose[{#, f /@ #}] &@(x /. Solve[f'[x] == 0, x])

Now, you can mark them on the plot

Plot[f[x], {x, -1.2, 1.2}, PlotRange -> {-0.12, 0.05}, 
 Epilog -> {PointSize[Medium], Point[zero], PointSize[Medium], Red, 
   Point[stationaryPoints]}, Frame -> True]

I hope this helps.