mth101 current gdb solution

Question:

f(x) = {x^3} - 5{x^2} + 3x + 1,
Find its first order derivative. 
Find its critical points in the interval [0,1]. 
Find its functional values at critical points and at end points of the interval [0,1].
Determine whether its absolute maximum and absolute minimum values occur at critical points or at the end points of the interval [0,1].

Answer

The first derivative of the function f(x), which we write as f 0 (x) or as df dx , is the slope of the tangent line to the function at the point x. To put this in non-graphical terms, the first derivative tells us how whether a function is increasing or decreasing, and by how much it is increasing or decreasing. This information is reflected in the graph of a function by the slope of the tangent line to a point on the graph, which is sometimes describe as the slope of the function. Positive slope tells us that, as x increases, f(x) also increases. Negative slope tells us that, as x increases, f(x) decreases. Zero slope does not tell us anything in particular: the function may be increasing, decreasing, or at a local maximum or a local minimum at that point.

to find these critical points you must first take the derivative of the function. Second, set that derivative equal to 0 and solve for x. Each x value you find is known as a critical number. Third, plug each critical number into the original equation to obtain your y values.