The maximum height the considered object will reach with the given facts is when t = 8 seconds.
To find the maximum of a continuous and twice differentiable function f(x), we can firstly differentiate it with respect to x and equating it to 0 will give us critical points.
Putting those values of x in the second rate of function, if results in negative output, then at that point, there is maxima. If the output is positive then its minima and if its 0, then we will have to find the third derivative (if it exists) and so on.
We're given that:
The height of the object is given as a function of time as:
[tex]h = -5t^2 + 80t[/tex]
where h is in meters and t is in seconds.
The function is polynomial, and therefore infinitely differentiable (a property of polynomials).
Finding the first and second derivative of height function with respect to time 't', we get:
[tex]h' = -10t + 80\\h'' = -10 < 0[/tex]
Putting second rate = 0 to find the critical values:
[tex]h'=0 \\-10t + 80 = 0\\t = 8 \: \rm seconds[/tex]
t = 8 is the only crticial value obtained.
Since the second rate is negative, all the critical values correspond to being local maxima.
Since there is only one critical value, and it is maxima, it is global maxima.
Thus, the height function will reach to the greatest value when t = 8 seconds.
Thus, the maximum height the considered object will reach with the given facts is when t = 8 seconds.
Learn more about maxima and minima of a function here:
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