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# Practice Paper 1 Question 6

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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## Hints

• Hint 1
How do you find points of maxima/minima of a given curve?
• Hint 2
What is the value of derivative of $$f(x)$$ at $$x=-1$$?
• Hint 3
How do you determine if a stationary point is a point of maxima?

## Solution

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 $$k<-\frac{4}{3}$$.

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