# Tag Info

### Show that $f(x)=\cos x-\sin x$ is bounded

Never subtract two inequalities which are in the same direction It some time you get right result and wrong other time. So subtraction of inequalities is a spurious exercise as the subtraction is non-...
• 39.3k

### Show that $f(x)=\cos x-\sin x$ is bounded

Just apply the triangle inequality: $$\lvert \cos x-\sin x\rvert\leq\lvert \cos x\rvert +\lvert \sin x\rvert\leq 2.$$
• 3,521
1 vote

### Show that $f(x)=\cos x-\sin x$ is bounded

You can write $$\begin{cases}-1\le\cos x\le1\\{-1}\le\sin x\le1\end{cases}\\ \implies \begin{cases}-1\le\cos x\le1\\{1}\ge-\sin x\ge -1\end{cases}$$ We see that it is not possible to sum these ...
• 973

• 343k

### Smallest value of $b$ when $0<\left\lvert \frac{a}{b}-\frac{3}{5}\right\rvert\leq\frac{1}{150}$

HINT: If $\frac{a}{b}$ satisfies the condition then $$|5 a - 3 b| = 1$$ the explanation being that in the parallelogram on the vectors $(3,5)$ and $(a,b)$ there should be no other integer points ...
• 47.8k

### Finding the minimum value of the expression $xy(x+y)^2 + yz(y+z)^2 + zx(z+x)^2$

Note that, If $x,y\to 0$ with $z\to 1$, we have $$\inf \left\{xy(x+y)^2 + yz(y+z)^2 + zx(z+x)^2\mid x+y+z=1 \wedge x>0\wedge y>0\wedge z>0\right\}=0$$ As for obtaining the inequality $A≥4xyz$,...
• 973
Accepted

• 39.3k
1 vote
Accepted

### Confusion regarding a continuous function achieving its minimum.

The $(a_j,b_j)$ don't cover the whole domain, they just consist of the points where $f>\lambda$. The point then is that $f$ must equal $\lambda$ at the endpoints of these intervals, otherwise ...
• 94.9k
1 vote

### Find the maximum attained by $f(x,y)=\frac{|a_1+a_2x+a_3(y-\alpha)|}{\sqrt y}$ in the triangle $\mathcal T$

Forming the lagrangian $$L(x,y,\lambda,s) = \frac{1}{y}(a_1+a_2 x+a_3(y-\alpha))^2+\lambda_1(x-s_1^2)+\lambda_2(x-y+\alpha+s_2^2)+\lambda_3(y-\alpha-1+s_3^2)$$ here $s_1,s_2,s_3$ are slack variables ...
• 26.5k
1 vote

### Determine the points in $[0, π]$ at which $f(x) = e^{2 \sin x−x}$ attains its maximum and minimum values.

So we want to find the maximum and minimum values of the function $$f(x)=e^{2\sin(x)-x}.$$ A differentiable function $f$ has a maximum (or minimum) if $f'=0$. Since $e^x$ and $2\sin(x)-x$ are ...
• 436
### Determine the points in $[0, π]$ at which $f(x) = e^{2 \sin x−x}$ attains its maximum and minimum values.
Once you differentiate $f(x)$ and find $f'(x)=(2\cos(x)-1)e^{2\sin(x)-x}$, you should determine where $f'(x)$ is positive. The inequality $(2\cos(x)-1)e^{2\sin(x)-x}> 0$ can be re-written as \$2\cos(...