Video: Finding the Monotonicity of a Function given Its Derivative Graph

The graph of the derivative 𝑓′ of a function 𝑓 is shown. On what intervals is 𝑓 increasing or decreasing?

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Video Transcript

The graph of the derivative 𝑓 prime of a function 𝑓 is shown. On what intervals is 𝑓 increasing or decreasing?

To answer this question, we need to recall the link between whether a function is increasing or decreasing and its first derivative. Formally, a function is increasing on an interval 𝐼 if 𝑓 of 𝑥 one is less than 𝑓 of 𝑥 two for all pairs of 𝑥-values, 𝑥 one and 𝑥 two, with 𝑥 one less than 𝑥 two in the interval 𝐼. In practical terms though, this just means that the graph of the function is sloping upwards. And so its first derivative which, remember, is the slope function of the curve is positive. On the other hand, a function is decreasing on an interval 𝐼 if 𝑓 of 𝑥 one is greater than 𝑓 of 𝑥 two for all 𝑥 one less than 𝑥 two in the interval 𝐼, which in practical terms just means the line is sloping downwards. And so the first derivative, 𝑓 prime of 𝑥, is negative.

To determine the intervals on which any function is increasing or decreasing then, we just need to consider the sign of its first derivative. So the function 𝑓 will be increasing when the graph of its first derivative 𝑓 prime is above the 𝑥-axis. From the given figure, we see that this is true on the open interval, one to five. 𝑓 will be decreasing when the graph of its first derivative is below the 𝑥-axis. From the figure, we see that this is true on two open intervals, the interval zero, one and the interval five, six. So we can conclude then that 𝑓 is increasing on the open interval one to five and decreasing on the open intervals zero to one and five to six.

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