I need help in this please !!! Can anyone help me in absolute values .

Question

Find the absolute maximum and absolute minimum values of the functionf(x)=5x^7−7x^5−7

on the interval [−4,4].

Enter -1000 for any absolute extrema that does not exist.

Absolute maximum = ... ?

Absolute minimum = ...?

Answer 1

To evaluate extrema (absolute maxima and absolute minima), you first have to be sure the function is continuous on the interval to be evaluated, and then assess the y values for all critical x-values. The critical values are the boundaries of the interval and each x value where the slope of the curve equals zero. Those points are where the curve changes direction from up to down (a maximum) or from down to up (a minimum). An example would be the vertex of a parabola.

First, this function is continuous since it is a polynomial. Don't let the 7th power worry you, it is still a polynomial, and therefore continuous.

Next, find the critical x values. The boundaries are -4 and 4, so those must be evaluated. To find the other critical x values, you need to determine where the slope is equal to zero. Those would be the points where the first derivative (which is the slope of a function) is equal to zero.

So for f(x) = 5x⁷ - 7x⁵ - 7

f'(x) = 35x⁶ - 35x⁴

That factors to f'(x) = 35(x⁶ - 35x⁴), which in turn factors to:

f'(x) = 35 (x⁴) (x² - 1), and since we have the difference of two squares, that factors to:

f'(x) = 35 (x⁴) (x- 1) (x +1)

The values for x where f'(x) = 0 are 0, 1 and -1, so these x values also have to be assessed. After comparing the corresponding y values in the original function, f(x), the x value that has the highest y value is the maximum, and the x value that has the lowest y value is the minimum.

x y

4 74795

1 -4

0 -7

-1 -5

-4 -74759

From this is it clear that the highest and lowest y values for this interval are at the boundaries, so those would be the absolute maximun and absolute minimum

Hope this helps