Definition: Let f:[a,b] to R be continuous and suppose that f(a)< 0<f(b). Then t
ID: 2943120 • Letter: D
Question
Definition: Let f:[a,b] to R be continuous and suppose that f(a)< 0<f(b). Then there exists a point c in (a,b) such that f(c)=0. (Intermediate value Theorem) Suppose that f:[a,b] to R is continous. Then f has the intermediate property on [a,b]. That is, if k is any value between f(a) and f(b) [i.e.,f(a)<k<f(b) or f(b)<k<f(a)], then there exists a c in (a,b) such that f(c)=k. Show that the equation 3^x=5x^3 has at least one real solution. This is a natural log function equal to a monomial function(polynomial term).Explanation / Answer
Definition: Let f:[a,b] to R be continuous and suppose that f(a)< 0<f(b). Then there exists a point c in (a,b) such that f(c)=0. (Intermediate value Theorem) Suppose that f:[a,b] to R is continous. Then f has the intermediate property on [a,b]. That is, if k is any value between f(a) and f(b) [i.e.,f(a)<k<f(b) or f(b)<k<f(a)], then there exists a c in (a,b) such that f(c)=k. Show that the equation 3^x=5x^3 has at least one real solution. This is a natural log function equal to a monomial function(polynomial term).
CONSIDER THE FUNCTION
F[X] = 5[X^3] - [3^X]..
THIS IS OBVIOUSLY CONTINUOUS IN THE INTERVAL X=0 TO 1
AT X=0 WE HAVE
F[0] = 5*0-1 = -1 .................1
AT X=1 , WE HAVE
F[1]=5-3=2.........................2
SO FOR THE CONTINUOUS FUNCTION
F[X] = 5[X^3] - [3^X]...... , WE HAVE
F[0]=-1<0<2=F[1]
HENCE THERE IS A POINT X = C IN THE INTERVAL BETWEEN X=0 AND X=1 , WHERE
F[X]=0
THAT IS
F[X] = 5[X^3] - [3^X] = 0
THAT IS
5[X^3] = [3^X]..................
SO the equation 3^x=5x^3 has at least one real solution
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