Question

In Exercises \(15-22,\) use analytic methods to find (a) the local ex- trema, (b) the intervals on which the function is increasing, and (c) the intervals on which the function is decreasing. $$ f(x)=5 x-x^{2} $$

Short answer

The local maximum is at \( x = 5 / 2, \) the function is increasing on the interval \( (-\infty, 5 / 2), \) and decreasing on the interval \( (5 / 2, \infty).

Step-by-step solution

Differentiate the function

First take the derivative of \( f(x) = 5x - x^2 \) using the power rule for differentiation. This gives \( f'(x) = 5 - 2x \).

Find the critical points

Set the derivative equal to zero and solve for x. This gives \( x = 5 / 2 \). This is the only critical point and it occurs at \( x = 5 / 2 \).

Determine where the function is increasing and decreasing

Next evaluate the derivative \( f'(x) = 5 - 2x \) at points in the intervals \( (-\infty, 5/2) \) and \( (5/2, \infty) \) to determine where the function is increasing and decreasing. For the interval \( (-\infty, 5/2), \) if we take, say, \( x = 0 \), we get \( f'(0) = 5 \) which is greater than 0 which indicates that the original function is increasing on this interval. Similarly for the interval \( (5/2, \infty), \) if we take, say, \( x = 3 \) we get \( f'(3) = -1 \) which is less than 0 which indicates that our original function is decreasing on this interval.

Identify the local maxima and minima

As the function transitions from increasing to decreasing at \( x = 5 / 2, \) this point is a local maximum.