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5.4 Maximum and Minimum Points

Local Maximum and Minimum Points

  • A local maximum point is when immediately to the left of the point, the gradient is positive, and immediately to right the gradient is negative. The table at the right summarises this.

Put simply, it is a point that has y-value larger than its surroundings.

  • A local minimum point is when immediately to the left of the point, the gradient is negative, and immediately to right the gradient is positive. The table at the right summarises this.
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5.3 First Derivative and Function Characteristics

Increasing and Decreasing Function

  • f(x) is strictly increasing in an interval (a,b) if f^{\prime}(x)>0 for all values of x in that interval.
    In this case, the tangent to any points in the interval has a positive gradient.
  • Similarly, f(x) is strictly decreasing in an interval if f^{\prime}(x)<0.
    The tangent has a negative gradient.
  • If f(x) is said to be strictly increasing (or decreasing) without any interval specified, then f^{\prime}(x)\geq0 (or f^{\prime}(x)\leq0)for ANY values of x (or precisely the domain of f(x)).
    Note: Notice the signs here are \geq, \leq instead of >, <.
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2.2 Graphs of Antiderivative Functions

Deducing Antiderivative Function Graphs

Method 1

i) The sign of f^{\prime}(x) ( >0 or <0) decides the sloping (upward or downward) of f(x).

ii) Any x-intercept of f^{\prime}(x) (i.e. f^{\prime}(x)=0) shows the stationary point of f(x). If f^{\prime}(x) changes sign before and after passing the x-axis, then it is a turning point for f(x). If not, it is just a stationary point.

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2.1 Graphs of Derivative Functions

Deducing Derivative Function Graphs

  • Here, it is assumed that the function of the original function is unknown, only its graph.
  • Given the graph of a function, some facts can be told about the derivative, recalling the fact that the derivative function represents the instantaneous rate of change of the original function at different points.
  • Firstly, it is obvious that the original function and its derivative would usually have the same domain, except when certain x values (or range) where the function is not differentiable.
  • It is also logical that the graph of derivative follows that of the original. For instance, if the original function is a polynomial, so will the derivative, thus its graph would look like one.
  • We can then look at specific points. If there is(are) stationary point(s) found, we know that f^{\prime}(x)=0 at these point(s). This is because the graph f(x) at these points are somewhat horizontal, thus it’s instantaneous rate of change is 0.
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