Get Exploration Concavity And The 2nd Derivative Test For

The second derivative measures the instantaneous rate of change of the first derivative. The sign of the second derivative tells us whether the slope of the tangent line to is increasing or decreasing.

The Second Derivative Test relates to the First Derivative Test in the following way. If f″(c)>0, then the graph is concave up at a critical point c and f′ itself is growing. Since f′(c)=0 and f′ is growing at c, then it must go from negative to positive at c.

The biggest difference is that the first derivative test always determines whether a function has a local maximum, a local minimum, or neither; however, the second derivative test fails to yield a conclusion when y'' is zero at a critical value.

The second derivative test uses the first and second derivative of a function to determine relative maximums and relative minimums of a function.

The second derivative test states that if f(x) has a stationary point when x=c and if f″(c)=0 then the test is inconclusive and we should study the sign of the first derivative, f′(x), to classify the stationary point.

Working Definition. An inflection point is a point on the graph where the second derivative changes sign. In order for the second derivative to change signs, it must either be zero or be undefined. So to find the inflection points of a function we only need to check the points where f′′(x) is 0 or undefined.