y1 differentiation

Question 1

conditions for minimum point

Answer 1

Gradient (f′(x)) changes from positive → 0 → negative.

Question 2

conditions for maximum point

Answer 2

Gradient (f′(x)) changes from negative → 0 → positive

Question 3

conditions for point of inflection

Answer 3

The graph doesn’t go from up to down or down to up.

Instead, it changes its curvature:

From “smiley face” (concave up) to “frowny face” (concave down), or vice versa.

The slope doesn’t have to be zero — it could keep increasing or decreasing.

Second derivative changes sign. (That’s the key.)

📌 Example:
f(x) = x^3
f′(x) = 3x²

f″(x) = 6x

At x = 0, f″(0) = 0, and it changes sign → point of inflection