SOLUTION OF PROBLEM: 8

Prove that the f(x) = 2(x)cube + 3(x)sq+6x +10 will always be an increasing function. (DC/M)

Solution: If we differentiate the given function we will get f'(x) = 6(x)sq + 6x +6 = 6( (x)sq +x +1)

Now, we can write f'(x) = 6[(x + 1/2)sq +3/4] which is always positive.So f(x) will always be an increasing function.

Alternatively, we can also say the for the quadratic expression (x)sq +x +1 coefficient of (x)sq is positive and discriminant is negative (-3), so the expression is always positive