V is the set of all real-valued functions defined and continuous on the closed interval [0,1] over the real field.
Prove/disapprove whether the set of all functions W belonging to V, which has a local extrema at x=1/2, is a vector space or not.
P.s : I am confused at second derivative test, as the linear combinations of 2 functions may result in a function whose second derivative at x=1/2 is zero.
Best Answer
Let $f(x)=(x-1)(2x-1)^2$ and $g(x)=x(2x-1)^2$.
Then $f^{\prime}(x)=(2x-1)(6x-5)$ and $g^{\prime}(x)=(2x-1)(6x-1)$,
so $f$ and $g$ are both in W $\;\;$(since f has a local max. at $\frac{1}{2}$ and g has a local min. at $\frac{1}{2}$).
However, $f+g$ is not in W,
since $(f+g)(x)=(2x-1)^3$ and $(f+g)^{\prime}(x)=6(2x-1)^2,\;$ so $f+g$ is always increasing.
Therefore W is not a subspace of V.