MediumMathematicsHigh School (11–12)Functions and Calculus › Application of DerivativesNumericalA-LevelAPIBJEE Main
Let f(x) = x³ - 6x² + 12x - 5. Find the critical points of f…
Contributed by StemPy LabTry it on paper first. A hint, the key idea and the worked solution are here when you need them.
Let . Find the critical points of . Decide whether each is a local maximum, a local minimum or neither, and give the number of local extreme values of . Find also the value of at the critical point.
Need a reminder? The key ideaShow
A critical point is a value where or does not exist. If changes from to at there is a local maximum, and from to a local minimum. If keeps its sign, there is neither.
The worked solution
Have you tried it?
Work it out on paper first. It is fine to get stuck or make a mistake: that is how we learn. When you are ready, go through the solution one step at a time, and check each step against your own.
Step 1. The derivative. .
Step 2. The critical point. only at . There is critical point.
Step 3. The signs. on both sides, and it is only at . So does not change sign.
Step 4. The conclusion. The point is neither a maximum nor a minimum. There are local extreme values. The function is strictly increasing, with a flat moment at .
Step 5. The value. .
Check. The second derivative is at , so the second derivative test would have told us nothing.
Answer: critical point, local extreme values, and .
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