Question

50 points: Please include how to find the derivative to solve the integral

Answer 1

the max value is where the sign of the deritivive changes from positive to negative

well, we see that the sign changes from positive to negative at x=6
so yah, find f(6), how do we do that?
no idea
well, the deritivive tells the slope at that point, so maybe we can use the slope to do something to tell us if the point is above or below the line
we find the slope of the line tangent at x=2
the slope at that point is about 16, and we have (2,10) s0
g(x)-10=16(x-2)
g(x)=16x-22
see that the deritive is positive the wohle time , so therefor the real value of f(6) is greater than g(6)
so find g(6)
g(6)=16(6)-22
g(6)=96-22
g(6)=74
so the value of g(6) has to be below that of f(x)

find the choice that is greater than 74

answer is D. 90

Answer 2

The maximum of
`f(x)` occurs at the value of
`x` for which
`f'(x)=0` and the sign of the derivative is positive to the left of
`x` and negative to the right. According to the graph of
`f'(x)`, this is the case for
`x=6`: you have
`f'(6)=0`, and
`f'(x)>0` for
`x` immediately to the left of 6, and negative immediately to the right.

By the fundamental theorem of calculus, you have


`\displaystyle\int_2^6f'(x)\,\mathrm dx=f(6)-f(2)`

which means


`\displaystyle f(6)=10+\int_2^6f'(x)\,\mathrm dx`

You can approximate the integral by finding the area under the curve for the interval
`[2,6]`, which can be done by counting the squares. I count 13 full squares right away, and maybe 5 or 6 additional ones from the remaining pieces. Each square has area 5, so the area is approximately 90 or 95.

So, you have


`f(6)\approx10+90=100`

or


`f(6)\approx10+95=105`

If A-D are the only options, that leaves D as the answer.