22. Relative Extrema

Dated: 30-10-2024

Relative Maxima

Most of the graphs we have seen have ups and downs.
the ups or the hills are called relative maxima.

Definition

A function¹ \(f(x)\) is said to have a relative maxima at \(x_0\) if \(f(x_0) \ge f(x)\) for all \(x\) in some open interval² \((a, b)\) containing \(x_0\).

Relative Minima

The downs and valleys are called relative minima.

Definition

A function¹ \(f(x)\) is said to have a relative minima at \(x_0\) if \(f(x_0) \le f(x)\) for all \(x\) in some open interval² \((a, b)\) containing \(x_0\).

Critical Points

These are the points on the graph where there is a transition of function¹ changing its nature from increasing to decreasing or other way around.

Theorem

If \(f(x)\) has a relative extrema at \(x_0\) then \(f^{\prime}(x_0) = 0\) or \(f(x)\) is not differentiable at \(x_0\).

The points where \(f^{\prime}(x_0) = 0\) are called stationary points.

First Derivative Test

Relative Maxima

If \(f^{\prime}(x) > 0\) on an open interval² starting from \(x_0\) and extending to left (towards \(-\infty\)) \(f^{\prime}(x) < 0\) when extending to right (towards \(+\infty\)) then \(f(x)\) is said to have a relative maxima at \(x_0\).

Relative Minima

If \(f^{\prime}(x) < 0\) on an open interval² starting from \(x_0\) and extending to left (towards \(-\infty\)) and \(f^{\prime}(x) > 0\) when extending to right (towards \(+\infty\)) then \(f(x)\) is said to have a relative minima at \(x_0\).

Infliction point

If the sign of \(f^{\prime}(x)\) remains the same at left and right sides of \(x_0\) then it means that the graph did not had a relative extrema at that point.
This point is called infliction point.

Second Derivative Test

1. if \(f^{\prime \prime}(x) > 0\) then \(f(x)\) has relative minima at \(x_0\).
2. if \(f^{\prime \prime}(x) < 0\) then \(f(x)\) has relative maxima at \(x_0\).

Example

Imagine a function¹ defined as:

\[f(x) = x^4 - 2x^2\]

Then we differentiate³ it

\[f'(x) = 4x^3 - 4x\]

\[= 4x(x-1)(x+1)\]

Taking the second derivative, we get

\[f^{\prime \prime}(x) = 12x^2 - 4\]

Since we know that the critical points exist at points where \(f^{\prime}(x) = 0\).
Therefore, to find such points:

\[0 = 4x(x-1)(x+1)\]

When we work this out, we get 3 points which are:

\[x = 0, \pm1\]

Now we substitute these values of \(x\) in \(f^{\prime \prime}(x)\),

\[f^{\prime \prime}(0) = -4 < 0\]

\[f^{\prime \prime}(1) = 8 > 0\]

\[f^{\prime \prime}(-1) = 8 > 0\]

Hence, \(f(x)\) has relative minima at \(x = \pm 1\) and relative maxima at \(x = 0\).

Graphs of Polynomials

Sometimes, during engineering, it is hard to graph the functions.¹
But we still need to understand the nature of the function¹ such as relative extrema and concavity etc.
For the purposes, assume a polynomial function \(P(x)\). - Use \(P^{\prime}(x)\) to determine stationary points and intervals² of increase and decrease. - Use \(P^{\prime \prime}(x)\) to determine infliction points and intervals² where \(P(x)\) is concave up and concave down. - Plot all these points and x and y intercepts as well.

Example

\[P(x) = y = x^3 - 3x + 2\]

\[\frac{dy}{dx} = 3x^2 - 3 = 3(x-1)(x+1)\]

\[\frac{d^{2}y}{dx^{2}}=6x\]

Stationary Points

From the first derivative test, we can find the stationary points which are at \(x = \pm 1\).
Put these values of \(x\) in the original \(P(x)\) to get corresponding \(y\) values, which are \(4\) and \(0\).

Intercepts

When we try substituting \(x = 0\) in \(P(x)\) to find y intercept, we get \(y = 2\).
When we try substituting \(y = 0\) in \(P(x)\) to find x intercept, we get \(x = -2\) and \(x = 1\).

Plot the points and we get something like

Concavity

• \(P^{\prime \prime}(-1) = -6\) suggests a relative maxima at \((-1, 4)\).
• \(P^{\prime \prime}(1) = 6\) suggests a relative minima at \((1, 0)\).

By considering these facts in mind, we can sketch our graph (an approximation) as:

Graphs of Rational Functions

Imagine a function¹ defined as

\[f(x) = \frac{P(x)}{Q(x)} = \frac{x}{x - 2}\]

This function¹ becomes undefined when \(x = 2\).
The graph of this function¹ looks something like this.

Horizontal Asymptots

A line \(y = y_0\) is called horizontal asymptots for the graph of \(f(x)\) if either is true

• \(\lim_{x \to +\infty} f(x) = y_0\)
• \(\lim_{x \to -\infty} f(x) = y_0\)

In the graph of the function¹ above, \(\lim_{x \to + \infty} f(x) = \lim_{x \to - \infty} f(x) = 1\).

Vertical Asymptots

A line \(x = x_0\) is called vertical asymptots for the graph of \(f(x)\) if either is true

• \(\lim_{x \to x_0^-} f(x) = - \infty\)
• \(\lim_{x \to x_0^+} f(x) = + \infty\)

In the case of graph of the function¹ above, \(\lim_{x \to 2^-} f(x) = - \infty\) and \(\lim_{x \to 2^+} f(x) = - \infty\). Hence, the vertical Asymptots is \(2\).

References

Read more about notations and symbols.

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1. Read more about functions. ↩↩↩↩↩↩↩↩↩↩↩

2. Read more about intervals. ↩↩↩↩↩↩

3. Read more about differentiation. ↩