Question

Find the maximum and minimum and draw the graph of f(x) = 4x ² - 40x + 80, for x = [0,8].

Answer 1

f(x) = 4x ² - 40x + 80, for x = [0,8]

To get the point of maxima/minima we first find the first derivative (with respect to x) of the function

f'(x) = 8x - 40

Setting f'(x) = 0

8x = 40

x = 5

We get x = 5 as the critical value. To find out if this is the maxima or minima, we see the sign of second derivative of the function.

f"(x) = 8

Since f"(x) >0, we have a minima at x = 5. Note that for positive value of second derivative function has a minima at its critical point. We show this by putting x = 5 in the function

For x = 5, we have f(x) = 4*25 -40*5+80 = 100 - 200 + 80 = -20

To find the maximum value of the function we check the value of function at end points.

For x = 0, f(x) = 0-0+80 = 80

For x = 8, f(x) = 4*64 - 40*8 +80 = 16

Therefore we see that at x = 0 function has a maximum value of 80 and at the critical x = 5 function attains minimum value of -20

Now to make the graph of the function, we make use of the values of f(x) at x = 0, 5, 8 and the graph is shown below