Find radius of convergence for <munderover> &#x2211;<!-- ∑ --> <mrow class="MJX-TeXAtom-

Find two different sets of parametric equations for the rectangular equation
${\displaystyle y=6x-5}$

Prove that $\left(\sum _{j=0}^N{a}_j\right)\left(\sum _{j=0}^N{b}_j\right)=\sum _{j=0}^{2N}{c}_j$

Calculate:
${\int }_0^{\pi }\cos (x)\log ({\sin }^2(x)+1)dx$

Let ${\displaystyle P\left(x\right)=F\left(x\right)G\left(x\right)}$ and ${\displaystyle Q\left(x\right)=\frac{F\left(x\right)}{G\left(x\right)}}$, where F and G are the functions whose graphs are shown.
a) Find ${\displaystyle P^{\prime }\left(2\right)}$
b) Find ${\displaystyle Q^{\prime }\left(7\right)}$

Represent the plane curve by a vector-valued function. ${\displaystyle x^2+y^2=25}$

Find definite integral.
${\displaystyle {\int }_0^2{x}^2{(3x^3+1)}^{\frac{1}{3}}dx}$

I have given real numbers $x_1,x_2,y_1,y_2$ such that $x_1>x_2$ and $y_1<y_2$. The the claim is that there exists some $\lambda \in (0,1)$ such that $\lambda (x_1-x_2)+(1-\lambda )(y_1-y_2)=0$. In order to proof this, one needs ( at least in my opinion) the intermediate value theorem. But the intermediate value theorem does not hold in constructive mathematics (that is without the law of excluded middle; or constructive mathematics acts in intuitionistic logic). Is there any constructive way to show the above equation?