Which values of
$k$ give a maximum at $x=-1$ for $f(x)=(k+1)x^4-(3k+2)x^2-2kx$?
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How do you find points of maxima/minima of a given curve?
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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?
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First derivative of
$f(x)$with respect to$x$is$(4k+4)x^3-(6k+4)x-2k$, 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$k<-\frac{4}{3}$.