Interpreting the Sign of the First Derivative (Increasing and Decreasing Functions)

Key Questions

  • Answer:

    It will be increasing when the first derivative is positive.

    Explanation:

    Take the example of the function #f(x) = e^(x^2 - 1)#.

    The first derivative is given by #f'(x) = 2xe^(x^2 - 1)# (chain rule). We see that the derivative will go from increasing to decreasing or vice versa when #f'(x) = 0#, or when #x= 0#.

    Whenever you have a positive value of #x#, the derivative will be positive, therefore the function will be increasing on #{x|x> 0, x in RR}#.

    The graph confirms

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    Hopefully this helps!

  • Answer:

    If #x_0 < x_1# then #f(x_0) < f(x_1)#

    Explanation:

    The meaning is that you have a function with positive slope in every point of Dom.

    Starting from a #x_0# and move to right, the graph of function is moving up at the same time

Questions