SOLUTION: if there is a function f and we know that f''(x)<>0 (f''(x) is not equal to 0) what does it mean for the f'(x) ?? (12th form student from Greece)

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Question 1064333: if there is a function f and we know that f''(x)<>0 (f''(x) is not equal to 0) what does it mean for the f'(x) ?? (12th form student from Greece)
Found 2 solutions by rothauserc, ikleyn:
Answer by rothauserc(4718) About Me  (Show Source):
You can put this solution on YOUR website!
f''(x) not = 0, tells us that f'(x) is increasing or decreasing
:
1) f''(x) < 0, we know the slope of the tangent line (f'(x)) to f(x) is decreasing as x increases and we say that f(x) is concave down(graphically a parabola that curves downward)
:
2) f''(x) > 0, we know the slope of the tangent line (f'(x)) to f(x) is increasing as x increases and we say that f(x) is concave upward(graphically a parabola that curves upward)
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Answer by ikleyn(52790) About Me  (Show Source):
You can put this solution on YOUR website!
.
Take (use) a parabola (a quadratic function) as a model, and it will say you everything.

If f''(x) > 0 , it is like a parabola with the leading coef. > 0, i.e. opened upward.

If f''(x) < 0, it is like a parabola with the leading coef. < 0, i.e. opened downward.