X and Y intercept Question

Question

Can someone explain why this function has no x-intercept:

f(t)= e^t - t

For the y-intercept I got 1.

Answer 1

Firstt step is finding out what the graph looks like without a graph. The y-intercept is found by evaluating the function at t=0:

f(0)=e^0−0=1 So the y-intercept is (0,1).

Consider the function g(t)=e^t−t we need to next find the first derivative to see how it acts and where its extrema are.

Its derivative is g′(t)=et−1,which at t=0 it is a minimum extrema.

Next, we can look at he second derivative to see if it is always concave up, which would explain why it never touches the x-axis:

g′′(t)=et >0 for all t.

This means the function is always concave up. Therefore, at t=0 the function has a minimum. Since the minimum value of the function is 1, the function never reaches zero. Furthermore, because it is always concave up and its lowest value is above the x-axis, the function has no x-intercept.

P.S. graphing the function could show you this, but I know a graphing calculator is not always available (: