pleases

x + y = 29/4  means  y = 29/4 - x

 

Let f(x) = x + y^2  so f(x) = x + (29/4 - x)^2

 

Differentiate this with respect to x:   df(x)/dx = 1 - 2(29/4 - x)  or  df(x)/dx = 2x - 27/2

 

Set this to 0 to find the value of x at which the function is 0:   x_min = 27/4

 

Put this baxk into the function:   f(27/4) = 27/4 +(29/4 - 27/4)^2  or  f(27/4) = 7

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x+y=29/4,

x=29/4-y

 

find minimum of x+y^2

Let

\(S=x+y^2\\ S=\frac{29}{4}-y+y^2\\ \mbox{This is a concave up parabola so any turning point will be a minimum}\\ \frac{dS}{dy}=-1+2y\\ \mbox{Find turning point }S'=0\\ -1+2y=0\\ 2y=1\\ y= \frac{1}{2}\\ x=\frac{29}{4}-\frac{1}{2}\\ x=\frac{29}{4}-\frac{2}{4}\\ x=\frac{27}{4}\\ \)

 

\(\mbox{Minimum S} = \frac{27}{4}+(\frac{1}{2})^2\\ \mbox{Minimum S} = \frac{27}{4}+\frac{1}{4}\\ \mbox{Minimum S} = \frac{28}{4}\\ \mbox{Minimum S} = 7\\\)

 

Here is the graph only y is swapped for x and S is the y axis.

 

https://www.desmos.com/calculator/smlz9rtuu0