Why is the stationary solution showing different behaviour than that of plotting maximum and minimum separately?

I am trying to plot the stationary solution of the function given below w.r.t. a parameter c as follows

f2[x_] := x^4 - b x^2 + c x;
b := 1
c := 0.1
sol00 = Solve[(f2'[x] == 0), x];

Manipulate[LogLinearPlot[x /. sol00, {c, 0.1, 0.7}], {c, 0.1, 0.7}]

Stationary points
The stationary solution doesn’t vary with parameter c which I am expecting and is shown from the the FindMinimum and FindMaximum function as follows

FindMinimum[f2[x], {x, 0}];
LogLinearPlot[x /. Last[FindMinimum[f2[x], {{x, 0}}]], {c, 0.1, 0.7}]

Local minimum
See how it varies with c.

FindMaximum[f2[x], {x, 0}];
LogLinearPlot[x /. Last[FindMaximum[f2[x], {{x, 0}}]], {c, 0.1, 0.7}]

Local maximum
this too varying with the parameter c. How can I get the similar behaviour from the stationary solution? I am doing some mistakes here, Please correct me. Thanks