The main difference between the first and second derivative tests is in the application of these tests when finding the maxima and minima of a given function.

- The first derivative tests involves differentiating the function and setting the first derivative equal to zero. Solving this gives us some values for x (called critical values). The first derivative test tells us that the function attains either a local extrema at these critical values of x. We cannot decide whether the the local extrema is a maxima or a minima using the first derivative test.
- We use the second derivative test in order to decide whether the critical value is a maxima or a minima. If the second derivative at the critical point is positive then we have a minima at that point whereas if the second derivative at the point is negative then we have a maxima at that point. If the second derivative at the point happens to be zero then we cannot make any conclusion about the nature of the critical point.

**Thus we see that the main difference between the first and second derivative tests is that – the first derivative test tells us at which point the function attains a local extremum whereas the second derivative test tells us whether that local extremum is a local maximum or a local minimum.**

**Example:**

Consider the function given by the polynomial, f(x)=\frac{1}{3}x^3-\frac{3}{2}x^2+2x+1. The first and second derivatives of the function are: f'(x)=x^2-3x+2 f"(x)=2x-3

We first apply the first derivative test by setting, f'(x)=0 x^2-3x+2=0 (x-1)(x-2)=0 x=1\text{ OR }x=2 This means that the function has local extrema at the points x=1 and x=2.

We now apply the second derivative test as follows.

At x= 1 the second derivative is given as, f"(1)=2*1-3 = 2-3 = -1 <0 Since the second derivative is negative it means that the function has a local maxima at the point x=1.

At x= 2 the second derivative is given as, f"(2)=2*2-3 = 4-3 = 1> 0 Since the second derivative is positive it means that the function has a local minima at the point x=2.