Question

For the following function, find all its critical point(s) and its absolute extrema.
f(x)=10sqrt(x^2+10)−3x , 0<=x<=14
PLEASE HELP ! I need all the steps because I have no idea how to do this! Thanksss

Answer 1

1. calculate the derivative of f
2. find the roots of the derivative
3. check if those are local maxima, minima or saddle points, you could use the second derivative for that
4. compare the function values at the local extrema to the border points

Answer 2

can you show me the actual solving of the derivative part?
chain rule?

Answer 3

yes, just the basic rules, of course chain rule for the square root part, just try it on paper

Answer 4

Oh my goodness, I typed the solution for you earlier, did you see it? I cant see it here now

Answer 5

f'(x) = [50x/[sqrt(x^2+10)] - 3 = [10x - 3sqrt(x^2+10)]/[sqrt(x^2+10)]

By the definition of a critical number, the critical numbers can be obtained by solving 10x - 3sqrt(x^2+10) = 0 and sqrt(x^2+10) = 0

From the first equation we get x^2 = 90/91. So x = sqrt(90/91) because x is between 0 and 14.
From the second equation, note that there is no solution, so no critical number comes from here.

So the critical point is (x,f(x)). This critical point will give a local extremum or an inflection point. But to find the absolute extrema, we don't have to know whether (x,f(x)) is a local extremum or not, you just need to compare the values of f(0), f(14) and f(sqrt(90/91)) and choose the largest and smallest value.

Answer 6

nadeem you are right , thanks