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How many critical points exist for the function y = f(x) within the interval x ∈ [0, 2], if the function satisfies d²y/dx² = 6x - 4 and has a local minimum value of 5 at x = 1?

1. None
2. One
3. Two
4. Three

Correct answer: Two

Solution

From d2y/dx2 = 6x-4, dy/dx = 3x^2 - 4x + C; the local minimum at x=1 forces dy/dx(1)=0, giving C=1, so dy/dx = 3x^2-4x+1 = (3x-1)(x-1). This vanishes at x=1/3 and x=1, both in [0,2], so there are two critical points. Stored 'one' is wrong.

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