A graph is said to be concave up (upward) at a point if the tangent line to the graph at that point lies below the graph.
When the function y = f (x) is concave up, the graph of its derivative y = f '(x) is increasing, i.e.
the graph of y = f (x) is concave upward on those intervals where y = f "(x) > 0.
We have "f''(x)=e^x>0" , so the graph of f(x) = ex curves upward.
Tangent line for f(x):
"y-y_0=e^{x_0}(x-x_0)"
"g(x)=e^{x_0}(x-x_0)+y_0"
"x=(g(x)-y_0)\/e^{x_0}+x_0"
So, for larger values of x:
"x\\to x_0"
So, tangent line of f(x) looks like straight vertical line.
For larger values of x, the graph of f(x) = ex appears to shoot straight up with no curve, and it is not optical illusion.
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