Show that arctan(x)+e^x+x^3=0 has a unique solution.

Since either sketching the function f(x)=arctan(x)+ex+x3 or evaluating the precise/approximated solutions of the equation would be impossible with A-level techniques, we have to come up with an "alternative method": the derivative one. First of all, we easily notice that the domain of the function is R and that it is continous on R (since it is a sum of continous functions). The derivative, which gives us the slope of the function, is f'(x)=1/(1+x2)+ex+3x2.
Now, 1/(1+x2)>0 for all x and so is ex. 3x2 is >=0 but when x=0 f(0)=2 so the derivative is always greater than 0. As a corollary of Lagrange's theorem, positive derivative implies strictly increasing function. Being f(x) continous and being the limit to -inf of f(x) = - inf and limit to +inf of f(x) = +inf, we can show that the function intersect the x-axis only once (Bolzano's theorem); therefore the given equation has a unique solution.

Answered by Maths tutor

3209 Views

See similar Maths A Level tutors

Related Maths A Level answers

All answers ▸

Given y = x^3 + 4x + 1, find the value of dy/dx when x=3


Separate (9x^2 + 8x + 10)/(x^2 + 1)(x + 2) into partial fractions.


ABCD is a rectangle with sides of lengths x centimetres and (x − 2) centimetres.If the area of ABCD is less than 15 cm^2 , determine the range of possible values of x.


Mechanics (M1): Particle moving on a straight line with constant acceleration (Relationships of the 5 Key Formulae)


We're here to help

contact us iconContact ustelephone icon+44 (0) 203 773 6020
Facebook logoInstagram logoLinkedIn logo

© MyTutorWeb Ltd 2013–2025

Terms & Conditions|Privacy Policy
Cookie Preferences