Deratives and Curves

Question

Consider the equation below. (Round the answers to three decimal places. If you need to use -∞ or ∞, enter -INFINITY or INFINITY.)

fx) = e^8x + e^-x

^{I am having a hard time find the zeroes so I can check if F(x)i s increasing or decreasing and how would i find the concavity?}

^{I know the first derative is 8e^8x-e^-x}

^{and the Secound derative is 64e^8x+e^-x}

Answer 1

Angie: I agree with your first and second derivative calculation. If you set the first derivative equal to zero it crosses the X axis at X equal to -.231. The second derivative never crosses zero (x-axis). In fact the second derivative is greater than zero for all values of X. Now let's talk about the characteristics of the function. Where the second derivative is greater than zero, which is true in this case and the first derivative is equal to zero, the function has a minimum (in this case the minimum is at -.231). With respect to the critical point of -.231, the first derivative is negative which means the function itself is decreasing. Where the first derivative is greater than zero (at the critical point of -.231), this implies the function is increasing. I personally don't see an inflection point which is where the second derivative would cross the X axis which in this case it does not. So you can conclude that the function is concave up since it is decreasing to the left of the critical point and increasing to the right of the critical point.