Topic Content:
- Increasing & Decreasing Functions
The most common graphs name the input value x and the output value y, and we say y is a function of x, or y = f(x) when the function is named f.
The derivative of a function may be used to determine whether the function is increasing or decreasing at any interval in its domain. The information may be used to show a reasonably accurate sketch of the graph of the function.
The derivative of a function is positive over the range where it is increasing and negative where it is decreasing.
If y is increasing then, \( \frac{dy}{dx} \scriptsize > 0 \)
If y is decreasing then, \( \frac{dy}{dx} \scriptsize < 0 \)
Example 4.2.1:
Find the range of values of x for which y = x3 + 5x2 - 8x + 1 is increasing.
Solution:
Given y = x3 + 5x2 - 8x + 1
This equation is a cubic function.
Cubic graphs are graphs of a cubic function and can be recognised as they include a cubed term. e.g. x3 . Cubic graphs are s-shaped .
The derivative \( \frac{dy}{dx}\)of a cubic function (x3 ) will give us a
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