The diagram shows the graph of the derivative of a function f(x) for ` 0 le x le 4 ` with f(0) = 0. Which of the following could be correct statements for y = f(x)?

(a) Tangent line to y = f(x) at x = 0 makes an angle of ` sec^(-1) sqrt 5` with the x - axis.
(b) f is increasing in (0, 3).
(c) x = 1 is both an inflection point and the point of local extremum.
(d) Number of critical point on y = f(x) is two.

Explanation

Slope of y tangent at ` x = 0 is tan^(-1) (2) = sec^(-1) sqrt 5`.
Hence (a) is correct.
` f'(x) ge 0" for " x in (0, 3)`, hence f is increasing for (0, 3) , and so (b) is correct.
The x - axis is tangent to the curve at x = 1, therefore f''(1) = 0.
So x = 1 is point of inflection, but not the point of extremum as the sign of f'(x) does not change at x = 1.
Hence (c) is not correct.
Obviously, y = f(x) has two critical points as f'(1) = 0 and f'(3.5) = 0.