# h(t)=10/(1+7e^2t)-Fill in the blanksThe curve______(increases or decreases) without bond as t increase without bound. The curve is concave up for t<__ and concave down for t<___. (Please...

h(t)=10/(1+7e^2t)-Fill in the blanks

The curve______(increases or decreases) without bond as t increase without bound. The curve is concave up for t<__ and concave down for t<___. (Please round your decimal into three decial places)

what does this graph look like for the particular function given.

sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted on

You need to check if the function increases of decreases, hence, you should evaluate the derivative of the given function, using the quotient rule, such that:

`h'(t) = (10'*(1 + 7e^(2t)) - 10*(1 + 7e^(2t))')/((1 + 7e^(2t))^2)`

`h'(t) = (0 - 10*(14e^(2t)))/((1 + 7e^(2t))^2)`

`h'(t) = (140e^(2t))/((1 + 7e^(2t))^2)`

You need to notice that rests positive for all real t, hence, the function h(t) strictly increases over R.

You need to evaluate the second order derivative to check where the function is concave up or concave down.

`f''(x) = ((140e^(2t))'((1 + 7e^(2t))^2) - (140e^(2t))*((1 + 7e^(2t))^2)')/((1 + 7e^(2t))^4)`

`f''(x) = ((280e^(2t))((1 + 7e^(2t))^2) - 14*140e^(2t))/((1 + 7e^(2t))^4)`

`f''(x) = (280e^(2t) + 14*2*140*e^(2t) + 280*49*e^(4t) - 14*140e^(2t))/((1 + 7e^(2t))^4)`

`f''(x) = (8*280e^(2t) + 280*49*e^(4t))/((1 + 7e^(2t))^4)`

Notice that the second order derivative is positive for all real t, hence, the function h(t) is concave up over R.