Mathematics

Mathematics

Problem

Consider the function f(x)=x4−4x3+6x2+1, for all x∈R.f(x)=x^{4}-4x^{3}+6x^{2}+1,\text{ for all }x\in\mathbb R. Answer the following: (i) Obtain f′(x)f'(x). (ii) Find all critical points of ff. (iii) Using appropriate reasoning based on f′(x)f'(x), decide the nature of each critical point and hence determine every local maximum and local minimum value of ff. (iv) State the intervals on which ff is increasing and those on which it is decreasing.
Intuitive explanation

This problem is about understanding how a polynomial's derivative reveals its shape—where it peaks, valleys, and changes direction. By finding f′(x)f'(x) and solving f′(x)=0f'(x) = 0, you locate the critical points where the function pauses its climb or descent. The key insight is using the sign of f′(x)f'(x) around each critical point: if the derivative changes from positive to negative, you've found a local maximum; negative to positive means a local minimum. This sign analysis also tells you exactly where ff rises and falls.

Intuitive explanation

This problem is about understanding how a polynomial's derivative reveals its shape—where it peaks, valleys, and changes direction. By finding f′(x)f'(x) and solving f′(x)=0f'(x) = 0, you locate the critical points where the function pauses its climb or descent. The key insight is using the sign of f′(x)f'(x) around each critical point: if the derivative changes from positive to negative, you've found a local maximum; negative to positive means a local minimum. This sign analysis also tells you exactly where ff rises and falls.

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