No one will me! i need pros! tysm! for f(x) = 3 x 5 − 5 x 3 + 3 . determine the interval(s) on which f(x) is concave up and on which f(x) is concave down and indicate where f(x) has inflection point(s). (if there are none, tell me). also, indicate where f(x) has local maximum and local minimum points. also, can you me determine the interval(s) on which f(x) is increasing and on which f(x) is decreasing.

now, if you do a first-derivative test on those critical points, check the regions next to them, for example I checked x = 0.5 and x = -0.5, and both gave -2.8125, the value doesn't matter for the test, what matters is the sign, is negative, meaning on that region, the graph has a negative slope and thus is going downwards.

and then I checked x = 2 and x = -2, and both gave 180, which is positive, meaning the original graph is going up there, slope is increasing once you go passed the 1 or -1.

now, checking the inflection points when doing the second-derivative test, x = -1 gives -30, negative,    concave down
x = -0.5 gives 7.5, positive, concave up
x = 0.5 gives -7.5, negative, concave down
x = 1 gives 30, positive,  concave up

check the picture below, the arrows show the direction the slope going, and therefore how the original function is moving.

notice, before -1 is going up, reaches -1, then it goes down, meaning that's a peak, or maximum.

before 1 is going down, reaches 1 it dives, then goes back up, that's a minimum.

Answer

1 number

explanation

if we use a graphing calculator, there is only one such number that satisfies the conclusion of the mean value theorem, that is, a number such that

 

where the right-hand side is the average rate of change of f on the interval [0.5,0.75], which is the slope of the secant line.

in the attached graph, the black line is the secant line which has its slope as the average rate of change of f on [0.5,0.75], while the orange line is the only tangent line to the graph of f in the interval that satisfies the mvt conclusion.

The answer is, the streetlight is 18 inches tall.