Solve second derivative test involves the process differentiating the given algebraic function twice with respect to the given variable. Generally the derivative is discussed in calculus whereas it is mainly used to find the rate of change of the given function with respect to the change in the input. The following are the solved example problems with detailed step by step solution in second derivative to study for the test.
Second derivative test example problems to solve:
Example 1:
Solve the test function to find second derivative.
f(z) = 2z6 + 2 z5 + 3 z4 + 3z
Solution:
The given equation is
f(z) = 2z6 + 2 z5 + 3 z4 + 3z
The above function is differentiated with respect to z to find the first derivative
f '(z) = 2(6z 5) +2 (5 z4 ) +3(4 z3) + 3
By solving above terms
f '(z) = 12z 5 + 10z4 + 12 z3 – 3
The above function is again differentiated with respect to z to find second derivative
f ''(z) = 12(5z 4 ) – 10(4z3) + 12(3z2)
f ''(z) = 60z 4 – 40z3 +36z2 is the answer.
Example 2:
Solve the test function to find second derivative.
f(z) = 5z 2 +5z 4 + 12
Solution:
The given function is
f(z) = 5z 2 +5z 4 + 12
The above function is differentiated with respect to z to find the first derivative
f '(z) = 5(2z )+5(4 z 3 ) + 0
By solving above terms
f '(z) = 10z +20z3
The above function is again differentiated with respect to z to find second derivative
f ''(z) = 10(1 ) +20(3z2)
f ''(z) = 10 + 60z2 is the answer.
Example 3:
Solve the test function to find second derivative.
f(z) = 4z4 +5z 5 +6z 6 + 2z
Solution:
The given function is
f(z) = 4z4 +5z 5 +6z 6 + 2z
The above function is differentiated with respect to z to find the first derivative
f '(z) = 4(4z 3 )+5(5z 4 ) +6( 6z 5) +2
By solving above terms
f '(z) = 16z 3 +25z 4 +36 z 5 + 2
The above function is again differentiated with respect to z to find second derivative
f ''(z)= 16(3z 2) +25(4z 3) +36 (5z 4)
f ''(z)= 48z 2 +100z 3 +180z 4 is the answer.
Second derivative test practice problems to solve:
1) Solve the test function to find second derivative.
f(z) = z 3 + z 4 + z 5
Answer: f ''(z) = 6z +12z2+ 20z 3
2) Solve the test function to find second derivative.
f(z) = 2z 3+3z5 + 4z 6
Answer: f ''(z) = 12z + 60z3 + 120 z 4
Second derivative test example problems to solve:
Example 1:
Solve the test function to find second derivative.
f(z) = 2z6 + 2 z5 + 3 z4 + 3z
Solution:
The given equation is
f(z) = 2z6 + 2 z5 + 3 z4 + 3z
The above function is differentiated with respect to z to find the first derivative
f '(z) = 2(6z 5) +2 (5 z4 ) +3(4 z3) + 3
By solving above terms
f '(z) = 12z 5 + 10z4 + 12 z3 – 3
The above function is again differentiated with respect to z to find second derivative
f ''(z) = 12(5z 4 ) – 10(4z3) + 12(3z2)
f ''(z) = 60z 4 – 40z3 +36z2 is the answer.
Example 2:
Solve the test function to find second derivative.
f(z) = 5z 2 +5z 4 + 12
Solution:
The given function is
f(z) = 5z 2 +5z 4 + 12
The above function is differentiated with respect to z to find the first derivative
f '(z) = 5(2z )+5(4 z 3 ) + 0
By solving above terms
f '(z) = 10z +20z3
The above function is again differentiated with respect to z to find second derivative
f ''(z) = 10(1 ) +20(3z2)
f ''(z) = 10 + 60z2 is the answer.
Example 3:
Solve the test function to find second derivative.
f(z) = 4z4 +5z 5 +6z 6 + 2z
Solution:
The given function is
f(z) = 4z4 +5z 5 +6z 6 + 2z
The above function is differentiated with respect to z to find the first derivative
f '(z) = 4(4z 3 )+5(5z 4 ) +6( 6z 5) +2
By solving above terms
f '(z) = 16z 3 +25z 4 +36 z 5 + 2
The above function is again differentiated with respect to z to find second derivative
f ''(z)= 16(3z 2) +25(4z 3) +36 (5z 4)
f ''(z)= 48z 2 +100z 3 +180z 4 is the answer.
Second derivative test practice problems to solve:
1) Solve the test function to find second derivative.
f(z) = z 3 + z 4 + z 5
Answer: f ''(z) = 6z +12z2+ 20z 3
2) Solve the test function to find second derivative.
f(z) = 2z 3+3z5 + 4z 6
Answer: f ''(z) = 12z + 60z3 + 120 z 4
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