 Hello and welcome to the session. In this session we are going to discuss the following question and the question says that the function f of x is equal to 2 x raised to power 5 minus 5 x raised to power 4 minus 10 x raised to power 3 as critical points at x is equal to minus 1, x is equal to 0 and x is equal to 3 determine whether each of these critical points is the location of a maximum or minimum or a point of inflection. Before starting the solution of this question we should know some results. That is if x is equal to c is a critical point we consider the points c minus h and c plus h where h is very small taken in decimals then c minus h will lie to the left of c and c plus h will lie to the right of c. Then we calculate the value of the function f of x at x is equal to c minus h and x is equal to c plus h. If f of c minus h is less than f of c and f of c plus h is less than f of c then the function is maximum at x is equal to c and if f of c minus h is greater than f of c and f of c plus h is greater than f of c then f of x is minimum at x is equal to c. Also if f of c minus h is less than f of c and f of c plus h is greater than f of c or if f of c minus h is greater than f of c and f of c plus h is less than f of c then x is equal to c is the point of inflection. Now these results will work out as key idea for the solution of the given question. Now let us proceed to the solution. Now in this question we are given the critical points as x is equal to minus 1, x is equal to 0 and x is equal to 3. Now we have to check whether these points are maximum or minimum points or points of inflection. Now let us start with the critical value x is equal to minus 1. Now using the key idea we have c is equal to minus 1. Now let us take h as 0.1. Now we will find f of c minus h, f of c and f of c plus h. So we have c minus h as minus 1 minus of 0.1 that is equal to minus 1.1 and c plus h as minus 1 plus 0.1 which is equal to minus of 0.9. Now we find f of c minus h that is equal to f of minus 1.1 and this will be equal to Now we put the value of x as minus 1.1 in this function 2 into minus 1.1 whole raise to power 5 minus 5 into minus 1.1 whole raise to power 4 minus 10 into minus 1.1 whole raise to power 3. And using calculator we have found this value as 2.769. So f of c minus h that is equal to f of minus 1.1 is equal to 2.769. Now we find f of c that is equal to f of minus 1 and which is given by 2 into minus 1 whole raise to power 5 minus 5 into minus 1 whole raise to power 4 minus 10 into minus 1 whole raise to power 3 and this is equal to 2 into minus 1 minus 5 into 1 minus 10 into minus 1 and that is equal to minus 2 minus 5 plus 10 which is equal to 3. So f of c that is equal to f of minus 1 is equal to 3 also f of c plus h will be equal to f of minus 0.9 and this is equal to 2 into minus 0.9 whole raise to power 5 minus 5 into minus 0.9 whole raise to power 4 minus 10 into minus 0.9 whole raise to power 3. Now using calculator we have found this value as 2.829. So f of c plus h that is equal to f of minus 0.9 is equal to 2.829. So we see that f of c minus h is less than f of c since 2.769 is less than 3 and f of c plus h is less than f of c as 2.829 is less than 3. So the function is maximum here thus we can say f of x is maximum at x is equal to minus 1. Now we check for x is equal to 0. So we have c is equal to 0 and we take the value of h as 0.1. So we have c minus h as 0 minus 0.1 that is equal to minus 0.1 and c plus h as 0 plus 0.1 that is equal to 0.1. So now we will find f of c minus h and that will be equal to f of minus 0.1 to find the value of f of minus 0.1. Here we put the value of x as minus 0.1 and we have 2 into minus 0.1 whole raise to power 5 minus 5 into minus 0.1 whole raise to power 4 minus 10 into minus 0.1 whole raise to power 3. And using calculator we have found this value as 0.009 so f of c minus h which is equal to f of minus 0.1 is equal to 0.009. Now we find the value of f of c that is equal to f of 0 and this will be given by 2 into 0 raise to power 5 minus 5 into 0 raise to power 4 minus 10 into 0 raise to power 3. And this will be equal to 2 into 0 minus 5 into 0 minus 10 into 0 and this is equal to 0 minus 0 minus 0 which is equal to 0. So f of c which is equal to f of 0 is equal to 0 also f of c plus h will be equal to f of 0.1 and this is given by 2 into 0.1 raise to power 5 minus 5 into 0.1 raise to power 4 minus 10 into 0.1 whole raise to power 3. Now using calculator we have found this value as minus of 0.010 so f of c plus h which is equal to f of 0.1 is equal to minus of 0.010. So here we have f of c minus h is greater than f of c as 0.009 is greater than 0 and f of c plus h is less than f of c as minus of 0.010 is less than 0. So from the key idea it is the point of inflection plus x is equal to 0 is the point of inflection. Lastly we have the point x is equal to 3 so now we have c is equal to 3 and we take the value of h as 0.1 so we have c minus h that is 3 minus 0.1 is equal to 2.9 and c plus h that is equal to 3 plus 0.1 which is equal to 3.1. So f of c minus h will be equal to f of 2.9 and this will be given by 2 into 2.9 raise to power 5 minus 5 into 2.9 raise to power 4 minus 10 into 2.9 raise to power 3. Now using calculator we have found this value as minus of 187.308 so f of c minus h is equal to f of 2.9 that is equal to minus 187.308. Similarly f of c is equal to f of 3 that is given by 2 into 3 raise to power 5 minus 5 into 3 raise to power 4 minus 10 into 3 raise to power 3 and using calculator we have found this value as minus of 189 so f of c which is equal to f of 3 is equal to minus of 189 Now f of c plus h will be equal to f of 3.1 and that is equal to 2 into 3.1 raise to power 5 minus 5 into 3.1 raise to power 4 minus 10 into 3.1 raise to power 3 Now using calculator we have found this value as minus of 187.087 so f of c plus h is equal to f of 3.1 which is equal to minus of 187.087 So here we have f of c minus h is greater than f of c as minus of 187.308 is greater than minus of 189 and f of c plus h is greater than f of c as minus of 187.087 is greater than minus of 189 So from key idea it is the point of minimum thus f of x is minimum at x is equal to 3 thus f of x is equal to 2 x raise to power 5 minus 5 x raise to power 4 minus 10 x raise to power 3 is maximum at x is equal to minus 1 minimum at x is equal to 3 also x is equal to 0 is the point of inflection this is the required answer this completes our session hope you enjoyed this session