My question is how to prove that $f(x) = \sqrt x – 2x^2$ has its maximum at point $x_0 = \frac{1}{4}$
It is easy to do that by finding its derivative and setting it to be zero (this is how I got $x_0 = \frac{1}{4}$). But the task is to do that without using any calculus tools and I`m stuck at it.
My idea was to introduce $t = \sqrt x$ to get $f(t) = t – 2t^4$ and then find its maximum because I know how to do that with parabola which can be transformed to $a(x-x_0)^2 + y_0$. So I was trying to turn $f(t) = t – 2t^4$ into $f(t) = a(x-x_0)^4 + y_0$ but it seems impossible.
My second thought was to use the definition of rising function on an interval so I supposed we have $a, b \in \left(0; \frac{1}{4}\right) \text{and } a < b$. Then I prove that $$\sqrt a – 2a^2 < \sqrt b – 2b^2$$ $$2(b – a)(a + b) < \sqrt b – \sqrt a$$ $$ 2(\sqrt a + \sqrt b)(a + b) < 1$$ which is true since $0 < a < b < \frac{1}{4}$
Exactly the same way I prove that for every $a, b \in \left(\frac{1}{4}; +\infty\right) \text{and } a < b$ $$\sqrt a – 2a^2 > \sqrt b – 2b^2$$
So we have that $f(x)$ is rising on $\left(0; \frac{1}{4}\right)$ and declining on $\left(\frac{1}{4}; +\infty\right)$, consequently at $\frac{1}{4}$ we have a maximum of $f$
But I guess it is not fair to use derivatives to find the maximum and then simply prove that this value is correct. Is there an even better solution? What are your thoughts about my proof?
UPD: there are a lot of solutions which are formally OK but first we have to guess $x_0 = \frac{1}{4}$ or $y_0 = \frac{3}{8}$. My goal is to find a solution which does not rely on guessing and I think the inequality above is the best way to do that
Best Answer
$f\left(\frac{1}{4}\right)=\frac{3}{8}.$
We'll prove that we got a maximal value.
Indeed, we need to prove that $$\sqrt{x}-2x^2\leq\frac{3}{8},$$ which is true by AM-GM: $$2x^2+\frac{3}{8}=2x^2+3\cdot\frac{1}{8}\geq4\sqrt[4]{2x^2\left(\frac{1}{8}\right)^3}=\sqrt{x}.$$
We can get a value $\frac{3}{8}$ by the following way.
Let $$\max_{x\geq0}f=k.$$ Thus, $$\sqrt{x}-2x^2\leq k$$ or $$2x^2+k\geq\sqrt{x}.$$ Now, since for $x\geq0$ we have $$\sqrt{x}=\sqrt[4]{x^2},$$ we need four addends if we want to use AM-GM.
And indeed, by AM-GM $$2x^2+k=2x^2+3\cdot\frac{k}{3}\geq4\sqrt[4]{2x^2\left(\frac{k}{3}\right)^3}.$$ Id est, we need $$4\sqrt[4]{2x^2\left(\frac{k}{3}\right)^3}=\sqrt{x}$$ or $$512k^3=27$$ or $$k=\frac{3}{8}.$$ The equality occurs for $$2x^2=\frac{k}{3}$$ or $$x=\frac{1}{4},$$ which says that we got a maximal value.