What are the extrema of #g(x) = 5x-80?# on the interval #[-1,10]#?

Question

What are the extrema of  #g(x) = 5x-80?# on the interval #[-1,10]#?

William Abdalla · Accepted Answer

The local extrema are #x = -1# and #x = 10#

The extrema of a function can be found where the first derivative is equal to zero. In this case the function is a line, so the end points of the function in the designated range are the extrema, and the derivative is the slope of the line.

Minimum: #(-1,-85)# Maximum: #(10, -30)

Elijah Abram · Answer

undefined To find the extrema of $ g(x) = 5x - 80 $ on the interval $[-1, 10]$, we first need to find the critical points. The critical points occur where the derivative of the function is equal to zero or undefined. Taking the derivative of $ g(x) $ with respect to $ x $ yields $ g'(x) = 5 $. Since this derivative is constant, it's never undefined, so there are no points where $ g'(x) $ is undefined.

Next, we set $ g'(x) = 0 $ to find the critical points. Solving $ 5 = 0 $ gives no solutions, so there are no critical points within the interval $[-1, 10]$.

This means that there are no local extrema within the given interval. However, we can check the endpoints of the interval to determine if there are any global extrema.

$ g(-1) = 5(-1) - 80 = -85 $

$ g(10) = 5(10) - 80 = 20 $

So, the minimum value occurs at $ x = -1 $ and the maximum value occurs at $ x = 10 $. Thus, the global minimum is $-85$ and the global maximum is $20$.