$k$
give a maximum at $x=1$
for $f(x)=(k+1)x^4(3k+2)x^22kx$
?

How do you find points of maxima/minima of a given curve?

What is the value of derivative of
$f(x)$
at$x=1$
? 
How do you determine if a stationary point is a point of maxima?

First derivative of
$f(x)$
with respect to$x$
is$(4k+4)x^3(6k+4)x2k$
, and second derivative of$f(x)$
with respect to$x$
is$(12k+12)x^2(6k+4)$
. Notice that the first derivative is always$0$
at$x=1,$
it does not depend on$k$
. For maxima, the second derivative must be negative at$x=1$
which gives us$12k+12 (6k+4)<0,$
and hence$k<\frac{4}{3}$
.