Question 1181070: If g(x) is real valued function defined on set of real numbers satisfying following three conditions:
(a) g'(0) = 0 (derivative of g at zero is equal to zero)
(b) g'' (-1) > 0 (second derivative of g at -1 is more than zero)
(c) g''(x) < 0 if x lies in (0, 2) where (0, 2) os open interval
What will be grpah of g(x)?
Answer by greenestamps(13216)   (Show Source):
You can put this solution on YOUR website!
You can't tell what the whole graph is going to look like; you can only determine certain features of the graph.
(a) g'(0) = 0 (derivative of g at zero is equal to zero)
The instantaneous slope of the graph is 0 at x=0, which means the tangent to the graph at x=0 is horizontal. This might be a maximum or minimum; or it might be a point of inflection.
(b) g'' (-1) > 0 (second derivative of g at -1 is more than zero)
Second derivative greater than zero means the graph is concave up; so at x=-1 the graph is concave up.
(c) g''(x) < 0 if x lies in (0, 2) where (0, 2) is open interval
Second derivative less than zero means the graph is concave down; so the graph is concave down on the interval (0,2).
With the graph concave up at x=-1, the tangent to the graph horizontal at x=0, and the graph concave down on the interval (0,2), the simplest polynomial function will be a cubic polynomial with an inflection point at x=0.
The function g(x)=-x^3 satisfies all those conditions:
g'(x) = -3x^2; g'(0)=0
g''(x) = -6x; g''(-1)=6 which is greater than 0; and g''(x) is negative for all positive values of x, including the interval (0,2).
A graph....
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