These exercises involve local maxima and minima of...

These exercises involve local maxima and minima of polynomial functions. a. Graph the function $P(x)=(x-1)(x-3)(x-4)$ and find all local extrema, correct to the nearest tenth. b. Graph the function $$Q(x)=(x-1)(x-3)(x-4)+5$$ and use your answers to part (a) to find all local extrema, correct to the nearest tenth.

These exercises involve local maxima and minima of polynomial functions.
a. Graph the function $P(x)=(x-1)(x-3)(x-4)$ and find all local extrema, correct to the nearest tenth.
b. Graph the function $$Q(x)=(x-1)(x-3)(x-4)+5$$ and use your answers to part (a) to find all local extrema, correct to the nearest tenth.

Key Concepts

Function Transformations

Function transformations alter the appearance of the graph without changing the inherent shape of the function. A key transformation is the vertical shift, where a constant is added or subtracted from the function. This shifts the entire graph up or down, thereby altering the numerical values of local extrema while leaving the structure of the graph unchanged.

Graphing Functions

Graphing functions involves plotting the points of the function to visually understand its behavior, intercepts, turning points, and other features. This is particularly useful for identifying local and global extrema, as well as for gaining intuition about the dynamics of the function over its domain.

Polynomial Functions

Polynomial functions are expressions that consist of variables raised to non-negative integer powers and their coefficients. They have smooth, continuous graphs and exhibit characteristic end behavior based on their leading term. The degree of the polynomial governs the number of turning points the function can have, and these functions are widely studied for their algebraic and calculus properties.

Local Extrema

Local extrema refer to the points on a function’s graph where the function reaches a local maximum or minimum value relative to its surroundings. In analysis, these points are often found by locating the critical points where the derivative is zero or undefined. They are important in understanding the overall behavior of the function, such as in optimization problems.

Transcript

00:01 So, i have a cubic polynomial.

00:04 Clearly for this i can observe and tell the roots are x equal to 1, 3 and 4.

00:11 So if i call this as the x -axis and this as the y -axis, i'll say 1, 3 and 4.

00:18 These are the roots.

00:19 And if i see f of 0, f of 0 will be minus 12.

00:23 All right, so it will come down, come up like this, will raise up, go down, and then again increase like this.

00:31 So this root is 1 .0 and this point is 3 .0 and this point is 4 .0.

00:48 So also this point where it cuts the y -axis, that is 0.

00:54 Minus 12.

00:55 So clearly these are the roots and if i talk about the local extremer, so i have to check out p dash x.

01:05 P -dash x will come out to be.

01:07 So first of all, let us simplify this px.

01:11 On simplification it will be obtained as x cube minus 8x square plus 19x minus 12.

01:20 Correspondingly, the derivative function p -dash -x will be equal to 3x square minus 16x plus 19.

01:28 So for the local extrema, i have to tell this value equal to 0.

01:35 So, whatever the roots are there for this quadratic, that will be the points of extrema.

01:40 So from the figure i can see that this is the point of extrema and this is the point of extrema.

01:45 Now, so now if i check out the roots of this quadratic, i will have x is equal to one root will be 3 .54 and the other root will be 1 .78.

02:08 So clearly it is greater than 1...

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