how do i find values at a given interval on a function?

Question

a sample problem: estimate the largest/smallest values of f(x)=2^x+x^2 at [4,1]

Answer 1

f(x)=2^x+x^2 at [4,1]
Now differentiate f(x)=2^x+x^2
f'(x) = 2^x (log2) +2x
for Max or Min put f'(x) =0
2^x (log2) +2x = 0
Since in the interval [4,1] the above equation does not satisfy because both parts i.e 2^x (log2) and 2x are positive
Now check
f(1)= 2^1+1^2=2+1=3
and f(4)= 2^4+4^2
= 16 +16=32
Hence the Max value=32
and Minimum Value = 3 in the interval[4 1]

Answer 2

Deviate the function to get the slope. Where the slope is =0 you have a max or min point. To find out which kind of point it is you can derive the function one more time. You now have a value of how the slope is changing. If f''(x)<0 you have a max and if f''(x)>0 you have a min.

Answer 3

You should use diffentiation to get maxima and minumum. Read the differentiation in any calculus text book