Final answer:
To determine the absolute extrema of the function f(x) within the given interval, we first find the derivative to identify critical points, then compute f(x) for these points along with the endpoints of the interval to find the absolute maximum and minimum values.
Thus the corret opction is:c
Step-by-step explanation:
The question is asking for the absolute extrema of the function f(x) = ln(3x² - 5x - 10) over a closed interval [5.1, 7.8]. To find extrema, we first need to find the derivative of f(x) to determine critical points.
The derivative of f(x) with respect to x is f'(x) = (6x - 5)/(3x² - 5x - 10). Setting the derivative equal to zero and solving for x will give us critical points within the interval. Moreover, since ln(x) is undefined for x ≤ 0, we must ensure that 3x² - 5x - 10 > 0 for the values of x.
After checking the critical points and endpoints of the interval, we can evaluate f(x) at these values to find the absolute maximum and minimum.
The answer, therefore, would be the pair (x, f(x)) of the critical point or endpoint which gives the highest and lowest values of f(x) on the interval.
The complete question is:content loaded
Find the absolute extrema for the given function f(x) = ln(3x² - 5x - 10) on the interval [5.1, 7.8].
a) (7.14, 3.22)
b) (6.92, 3.54)
c) (5.78, 2.91)
d) (6.45, 3.86)