I need to find the critical points (maxima, minima and points of inflection) of the function $f(x)= \sin(x)+\ln(\frac{1}{4}x^2-\frac{1}{4}x+\frac{3}{2})$ over the interval $-3\leq x\leq5$.
To do this I need to find all values x such that $f'(x)=0$ or $f'(x)$ does not exist.
How can this be done in MMA? (N)Solve and Reduce don't seem to work.
This is for an introductory Calculus course so please don't tell me the answer, just how to find it. Thanks.
Note a critical point of $f(x)$ must also lie in it's domain. So for example $x = 0$ is not a critical point of $f(x) = 1/x$, even though $f'(0)$ doesn't exist.
We can exploit the M10 function FunctionDomain.
RealD[f_, x_] := Assuming[x ∈ Reals, FunctionExpand@D[f, x]]
CriticalPoints[f_, x_] := CriticalPoints[f, {x, -∞, ∞}]
CriticalPoints[f_, {x_, a_, b_}] :=
Reduce[RealD[f, x] == 0 && a <= x <= b, x, Reals] ||
Reduce[FunctionDomain[f, x, Method -> {Reduce -> False}] &&
!FunctionDomain[RealD[f, x], x, Method -> {Reduce -> False}] &&
a <= x <= b, x, Reals]
I didn't test this code too much, so maybe it needs improvement. Here are some examples:
CriticalPoints[Sin[x] + Log[x^2/4 - x/4 + 3/2], {x, -3, 5}] // N
x == -1.169 || x == 1.94892 || x == 4.32773
CriticalPoints[Abs[x], x]
x == 0
CriticalPoints[1/x, x]
False
Reduce don't work because there are infinite x which satisfy f'[x]==0. If you add the range of x, it will give the answer, like this
Reduce[f'[x] == 0 && -3 <= x <= 5, x]
FindRootnear the suspicious points to find them precisely $\endgroup$