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I am asked to verify if this function is decreasing or increasing:

f(x)=sqrt(2+5x)

I am using Maple 15 right now in my class and writing the following commands can't solve the problem.

f := x-> sqrt(2+5*x): solve(diff(f(x), x) > 0, x);

After Run; it is shown to me that :

Warning, solutions may have been lost

In fact, I expect to have some intervals, but it isn't done properly. What can I do? Thanks for any help!

marc_s
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Mikasa
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2 Answers2

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Solve doesn't work that good with inequalities. An example:

solve((sqrt(x^2))>0,x);
                     RealRange(Open(0), infinity)

So half the solutions are lost without even a warning. You're going to have to do the thinking yourself. I would calculate the root of the original function, determine the valid input range from that and calculate the roots of the derivative. Seeing as there are no roots, its sign is constant and just fill in a valid x to get the sign of the derivative for the entire valid input range.

solve(f(x)=0,x);


                                 -2/5
solve(diff(f(x),x)=0,x);

subs(x=0,diff(f(x),x));

                                   1/2
                                5 2
                                ------
                                  4

So positive sign means increasing

Origin
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The domain of sqrt(2+5*x) is all points x such that 2+5*x is not less than zero.

For this example, you can ask Maple whether it can tell that the derivative of sqrt(2+5*x) is greater than zero for all x in the domain. If the answer is true then it is increasing everywhere in its domain.

restart:

is( diff(sqrt(2+5*x),x) > 0 ) assuming 2+5*x >= 0;

                          true
acer
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