Buckle in — you’re in the front seat. As the track bends up beneath you (concave up, f″>0) it slams you down into the seat; as it crests over (concave down, f″<0) your stomach drops and you lift out of it — airtime; and for one instant at an inflection point you’re weightless-normal, exactly 1 g. The force you feel is f″. Press Ride — or drag along the track to look around.
The second derivative is nothing exotic — it’s just the derivative of the derivative. Watching what that means turns “concavity” from a vocabulary word into something you can read at a glance.
f″>0: the slope is increasing, so the curve bends up like a cup — it holds water. f″<0: the slope is decreasing, the curve bends down like a dome — it sheds water.
At a critical point where f′(c)=0: if f″(c)>0 it’s the bottom of a cup → min; if f″(c)<0 it’s the top of a dome → max.
If f″(c)=0 the test is silent — fall back to the first-derivative test.
The star of this lesson: the exact spot where the bend flips — concave up on one side, concave down on the other. On the coaster it’s the single instant you feel normal gravity.
Two traps live right here. A place where f″=0 but the bend never flips, and a critical point where the second-derivative test simply gives up:
For f(x)=x⁴, f″(x)=12x² is zero at x=0 — but it’s positive on both sides. The curve stays concave up (a cup the whole way). No sign change, so x=0 is not an inflection point.
For f(x)=x⁴ at x=0, both f′(0)=0 and f″(0)=0. The 2nd-derivative test is inconclusive. Fall back to the first-derivative test: f′ goes −→+, so it’s a genuine minimum.
Differentiate twice. f″>0 means f′ is climbing (slopes getting steeper upward); f″<0 means f′ is falling. The bend is just the trend of the slope.
Candidates are where f″=0 or f″ DNE. It’s only an inflection if the sign of f″ actually changes across it.
A curve can rise while concave down (going up, but slowing) or fall while concave up. f′ is direction; f″ is bend — they’re independent.
If f″(c)=0 at a critical point, the second-derivative test tells you nothing — the point could be a max, a min, or neither. Use the first-derivative test instead.
Concave up/down is the sign of f″, not f′. A graph can be increasing and concave down at the same time.
Only if the sign changes. \(f(x)=x^4\) has \(f''(0)=0\) yet stays concave up — no inflection.
If \(f''(c)=0\) the second-derivative test is inconclusive. Don’t call it a max or min — switch to the first-derivative test.
\(f''>0\) is concave up (holds water) → a min at a flat spot. \(f''<0\) is concave down → a max.