 Hi and welcome to the session. Let us discuss the following question. Question says, find the values of a and b such that the function defined by fx is equal to 5. If x is less than equal to 2, fx is equal to ax plus b. If x is greater than 2 and less than 10, fx is equal to 21. If x is greater than equal to 10 is a continuous function. First of all, let us understand that function f is continuous at x is equal to a. If function is defined at x is equal to a or we can say f a exists. Left hand side limit of a function is equal to right hand side limit of a function is equal to value of the function at x is equal to a. This is the key idea to solve the given question. Let us now start with the solution. We are given function f which is defined by fx is equal to 5. If x is less than equal to 2, fx is equal to ax plus b. If x is greater than 2 and less than 10, fx is equal to 21. If x is greater than equal to 10, clearly we can see function f is defined at all the real values less than 2 equal to 2, greater than 2 less than 10, greater than 10 and equal to 10. Or simply we can say function f is defined at every real number. First of all let us discuss continuity of the function for all the real values less than 2. We know fx is equal to 5 if x is less than equal to 2. Now this is a constant function and constant function is continuous at every real number. So given function f is continuous at every real number less than 2. Let us discuss continuity of the given function for all the values of x greater than 10. We are given fx is equal to 21 if x is greater than equal to 10. Again this is a constant function and constant function is continuous at every real number. So function f is continuous at every real number greater than 10. Now we have to find the values of a and b so that the given function f is continuous at every real number. Now let us discuss continuity of the function at x is equal to 2, at x is equal to 2. Left hand side limit of the function is given by limit of x tending to 2 minus fx. We know for x less than 2 fx is equal to 5. So we can write limit of x tending to 2 minus 5 which is equal to 5 only. Now let us find out right hand side limit of the function at x is equal to 2. This is given by limit of x tending to 2 plus fx which is further equal to limit of x tending to 2 plus ax plus b we know for x greater than 2 fx is equal to ax plus b. Now this limit is equal to 2a plus b. Now we know given function is continuous at x is equal to 2 only when these two limits coincide each other. So 2a plus b must be equal to 5. Let us name this expression as 1. Now let us discuss continuity of the function at x is equal to 10. Now at x is equal to 10. Left hand side limit of the function is given by limit of x tending to 10 minus fx which is equal to limit of x tending to 10 minus ax plus b which is further equal to 10a plus b. So left hand side limit of the function at x is equal to 10 is equal to 10a plus b. Now let us find out right hand side limit of the function at x is equal to 10. This is given by limit of x tending to 10 plus fx which is further equal to limit of x tending to 10 plus 21. We know for x greater than 10 fx is equal to 21 which is further equal to 21 only. Now function is continuous at x is equal to 10 only when these two limits coincide each other. So we can write 10a plus b is equal to 21. Let us name this expression as 2. Now from expression 1 we get b is equal to 5 minus 2a. Let us name this expression as 3. Now substituting this value of b in expression 2 we get 10a plus 5 minus 2a is equal to 21. Now 10a minus 2a is equal to 8a. So we can write 8a plus 5 is equal to 21. Now subtracting 5 from both the sides we get 8a is equal to 16. Now dividing both the sides by 8 we get a is equal to 2. Now we will find the value of b by substituting this value of a in expression 3. We get b is equal to 5 minus 2 multiplied by 2. This implies b is equal to 5 minus 4. We know 2 multiplied by 2 is equal to 4. Now this implies b is equal to 1. So the required values of a and b are a is equal to 2 and b is equal to 1. At these values of a and b function is continuous function. Now we can write fx is equal to 2x plus 1 if x is greater than 2 and less than 10. We were given fx is equal to ax plus b if x is greater than 2 and less than 10. Here we have substituted corresponding values of a and b. Now clearly we can see this is a polynomial function and polynomial function is continuous at every real number. So given function f is continuous at every real number between 2 and 10. So given function is continuous at every real number for value of a is equal to 2 and value of b is equal to 1. This is our required answer. This completes the session. This would be the solution. Take care and have a nice day.