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Question

Let f be the function defined for x >> 0 (x is greater or equal 0) with f(0)=5 and f', the first derivative of f, given by f'(x)=e^(-x/4) sin(x^2). The graph of y=f'(x) is shown as the graph -> http://img483.imageshack.us/img483/5387/42521098qj9.png

a) Use the graph of f' to determine whether the graph of f is concave up, concave down, or neither on the interval 1.7 < x < 1.9. Explain
b) On the interval 0 << x << 3 (<< is greater or equal), find the value of x at which f has an absolute max. Justify the answer.
c) Write an equation for the line tangent to the graph of f at x=2

Answer 1

It looks like f'(1.7)= 0 and f' is changing from + to - as it goes through x=1.7. Hence f(x) must have a max at f(1.7) and is thus on its way down. Thus it's is concave down between x= 1.7 and 1.9.

From the graph, f'(2) appears to be -.5.
So y = -.5x+b
When x = 0, f(x) = 5
So 5 = -.5(0)+b --> b=5
So y=-.5x+5