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First Derivative Test vs Second Derivative Test

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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.

  1. 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.
  2. 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.

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