Maths WorkshopMaths problems with full solutions

Intermediate Value Theorem

17 problems · Existence of solutions and of zeros.

Ex.25A fixed point for every continuous self-map of [0,1]
#fixed points #intermediate value theorem
Intermediate Value Theorem
High school
High school
Ex.27Locating the roots of a cubic
#intermediate value theorem #polynomial roots #bracketing
Intermediate Value Theorem
High school
High school
Ex.28How long is the drug effective?
#applications #intermediate value theorem #bracketing
Intermediate Value Theorem
High school
High school
Ex.55The monk on the mountain
#intermediate value theorem #classic puzzle
Intermediate Value Theorem
High school
High school
Ex.87Fixed point on [0,1]
#fixed point #IVT #auxiliary function
Intermediate Value Theorem
High school
High school
Ex.105Commuting maps on $[0,1]$ share a fixed point
#commuting maps #IVT #extreme value theorem #induction
Intermediate Value Theorem
L1
L1
Ex.110Continuous functions with f^2=1 on an interval
#IVT #sign #constant functions
Intermediate Value Theorem
High school
High school
Ex.316A cubic with exactly one real root
#intermediate value theorem #monotonicity
Intermediate Value Theorem
L1
L1
Ex.360Odd degree polynomials have a real root
#intermediate value theorem #polynomials #limits at infinity
Intermediate Value Theorem
L1
L1
Ex.392The bisection method in practice
#bisection #numerical methods #intermediate value theorem
Intermediate Value Theorem
L1
L1
Ex.720A continuous function with rational values is constant
#intermediate value theorem #rational numbers #proof by contradiction
Intermediate Value Theorem
L1
L1
Ex.736cos x = 1/x has infinitely many solutions
#intermediate value theorem #infinitely many solutions
Intermediate Value Theorem
L1
L1
Ex.737Counting solutions depending on a parameter
#intermediate value theorem #monotonicity #parameters
Intermediate Value Theorem
L1
L1
Ex.756tan x = x on each branch
#intermediate value theorem #tangent #monotonicity
Intermediate Value Theorem
L1
L1
Ex.1479Continuous injective functions are monotone
#intermediate value theorem #injectivity #monotonicity
Intermediate Value Theorem
L2
L2
Ex.1491How many solutions does ln x = ax have?
#intermediate value theorem #parameters #logarithm
Intermediate Value Theorem
L1
L1
Ex.1512The horizontal chord of length one half
#intermediate value theorem #chords #clever auxiliary function
Intermediate Value Theorem
L2
L2