1point) are the functions f, g, and h given below linearly independent? f(x)=e3x+cos(5x), g(x)=e3x−cos(5x), h(x)=cos(5x). if they are independent, enter all zeroes. if they are not linearly independent, find a nontrivial solution to the equation below. be sure you can justify your answer. 1 (e3x+cos(5x))+ -1 (e3x−cos(5x))+ 0 (cos(5x))=0.

Functions are linearly dependent (are not linearly independent.)

Step-by-step explanation:

Remember that two functions f(x), g(x) and h(x) are said linearly independent on an interval I if the only solution to the equation

is the trivial one: α = 0, β = 0, ω = 0. If they are not linearly independent, they are called linearly dependent.

Now, let f(x), g(x) and h(x) be the functions:

Then, letting α = 1, β= -1 and ω = -2, we see that:

Hence, the functions f(x), g(x) and h(x) are not linearly independent, or equivalently, are linearly dependent.

solution:

the given function is ,

      =

the three roots of this quadratic function is , 0 ,8 and 10.all the roots are real and rational.

if a quadratic equation is of the form ,

sum of the roots = a + b +c= 0+8+10=18=  

product of roots taken two at a time = a b + b c+ c a = 0+80+0=80 =  

product of roots = a b c = 0=  

there are 11 doors in a hall starting from 0 to 11.the entry in the hall is permitted through gate number 0 and exit is allowed through only gate number 8 and 10.so, three roots(0,8,10) of quadratic function is given by entry and exit route of this hall.