Find the maximum value of fx y z Inx y32 z22 on the curve

Find the maximum value of f(x, y, z) In(x) + y^3/2 - z^2/2 on the curve vector r(t) (e^t, t^2, squareroot t)

first follow condition to satisfy the curve in r(t)

x= e^t

y=t^2

z=sqrt(t)

put these in f(x,y,z) gives

=ln(e^t)+(t^2)^3/2-sqrt(t)^2/2

=t+t^3-t/2 where t>0 as sqrt(t) is defined

=t/2+t^2

now our problem becomes

f(t)=t/2+t^3

so we can\'t get the maximum value of this equation we can get only minimum value of the function which is 0 at t=0;