If somebody has a experience with polynomials.
How to set this Hermite function to have a general minimum where I want on $x$ axis, for example in $0.5$. Is it possible to be in analytic form $f[x,a]$? Point $a$ is minimum.
$f[x] = -(1/48) + (3 x^2)/16 - (5 x^4)/16 + (7 x^6)/48$
And the same thing with this Hermite function to have one of its roots where I want also in form $g[x,b]$? Point $b$ is one of zeros.
$g[x] = -(x/2) + x^3/2$
For Example I want to do this but without connecting two functions.
f[x] function is
Plot[1/8 - x^2/4 + x^4/8, {x, -1, 1}]
and I moved the maximum in point a=47/133
Plot[Piecewise[{{2423393/2916000 + (831383 x)/972000 - (760627 x^2)/ 972000 - (2352637 x^3)/2916000, x < 47/133}, {-(17689/79507) + (2494149 x)/318028 - (2388015 x^2)/ 159014 + (2352637 x^3)/318028, x > 47/133}}], {x, -1, 1}]
Function g[x] is
Plot[-(x/2) + x^3/2, {x, -1, 1}]
and I moved zero in point b=47/133
Plot[Piecewise[{{47/180 - (43 x)/90 - (133 x^2)/180, x < 47/133}, {47/86 - (90 x)/43 + (133 x^2)/86, x > 47/133}}], {x, -1, 1}]
How to do this without connecting two functions?