Calculus · Higher Tier
The Summit: Maxima & Minima
Finding where gradients vanish and hills become valleys
The Summit Seeker
Position
(0.00, 0.00)
Gradient dy/dx
-3.000
Status
Moving...
Logic Track
The Zero Rule
At the top of a hill or bottom of a valley, for one split second, you are completely flat. The gradient is exactly zero.
Visual Clue
If you drew a tangent line at that point, it would be perfectly horizontal. No rise, no fall — just flat.
Why It Matters
Finding where dy/dx = 0 tells us where the curve changes direction. These are called turning points or stationary points.
Dynamic Solver: Turning Point Calculator
Function: y = 1x² -6x + 8
Step 1: Differentiate
dy/dx = 2x -6
Step 2: Set equal to zero
2x -6 = 0
Step 3: Solve for x
2x = 6
x = 3.000
Step 4: Find y-coordinate
y = 1(3.000)² -6(3.000) + 8
y = -1.000
Step 5: Check nature
Coefficient of x² is 1 (positive)
Shape: ∪ (smiley)
Result: Minimum at (3.000, -1.000)
📝 Mark Scheme Checklist
✓ Differentiated correctly
✓ Set derivative equal to zero
✓ Solved for x
✓ Substituted to find y
✓ Stated nature (max or min)
✓ Gave final answer as coordinates
Syllabus Reference: Higher Tier 3.4 (C & D) — Determine gradients, rates of change, stationary points, turning points by differentiation. Distinguish between maxima and minima by considering graph shape.
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