\[
f(x)=\frac{x^4}{4}+\frac{(k-3)x^3}{3}-\frac{3kx^2}{2}+k
\]
\[
\text{where }k\text{ is a positive constant.}
\]
Determine the regions where \(f(x)\) is decreasing.
Try the question yourself first. If you get stuck, open the hints before using the full walkthrough.
Differentiate first, factorise the derivative, find the critical points, use the second derivative test to classify them, then use the sign of the factorised derivative to decide where \(f(x)\) is decreasing.
Step 1
Differentiate the function
Differentiate term by term. Which derivative is correct?
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Step 2
Factorise the derivative
Factor out the common factor, then factorise the quadratic. Which factorised form is correct?
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Step 3
Find the critical points
Set \(f'(x)=0\). What are the critical x-values?
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Step 4
Differentiate again
We need the second derivative for the second derivative test. Which expression is correct?
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Step 5
Apply the second derivative test
Use \(k>0\) when evaluating \(f''(x)\) at the critical points. Which classification is correct?
Step 6
Find where the derivative is negative
Since \(k>0\), the critical points are ordered \(-k<0<3\). On which intervals is \(f'(x)<0\)?