# 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.

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### 1 Answer

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.**