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Find the critical points of the function f(x)=x3−3x+1f(x) = x^3 - 3x + 1.
Jan 22, 2024
Find the critical points of the function f(x)=x3−3x+1f(x) = x^3 - 3x + 1.
Generated Graph
Solution by Steps
step 1
To find the critical points of the function f(x)=x3−3x+1 f(x) = x^3 - 3x + 1 , we need to find the values of x x where the first derivative f′(x) f'(x) is equal to zero or undefined
step 2
First, we calculate the derivative of f(x) f(x) . The derivative of x3 x^3 is 3x2 3x^2 , the derivative of −3x -3x is −3 -3 , and the derivative of a constant 1 1 is 0 0 . So, f′(x)=3x2−3 f'(x) = 3x^2 - 3
step 3
Set the derivative equal to zero to find the critical points: 3x2−3=0 3x^2 - 3 = 0
step 4
Factor out the common factor of 3 3 : 3(x2−1)=0 3(x^2 - 1) = 0
step 5
Set the factor x2−1 x^2 - 1 equal to zero: x2−1=0 x^2 - 1 = 0
step 6
Solve the equation x2−1=0 x^2 - 1 = 0 by factoring it as (x+1)(x−1)=0 (x + 1)(x - 1) = 0
step 7
Set each factor equal to zero and solve for x x : x+1=0 x + 1 = 0 gives x=−1 x = -1 , and x−1=0 x - 1 = 0 gives x=1 x = 1
Answer
The critical points of the function f(x)=x3−3x+1 f(x) = x^3 - 3x + 1 are x=−1 x = -1 and x=1 x = 1 .
Key Concept
Critical Points of a Function
Explanation
Critical points occur where the derivative of the function is zero or undefined. In this case, the critical points are found by setting the first derivative equal to zero and solving for x x .
What is the first derivative of the function f(x)=x3−3x+1f(x) = x^3 - 3x + 1
Generated Graph
Solution by Steps
step 1
asksia-II calculator input: Differentiate x3−3x+1x^3 - 3x + 1
step 2
The derivative of x3x^3 is 3x23x^2
step 3
The derivative of −3x-3x is −3-3
step 4
The derivative of a constant, 11, is 00
step 5
Combine the derivatives: 3x2−3+03x^2 - 3 + 0
step 6
Simplify the expression: 3x2−33x^2 - 3
Answer
The first derivative of the function f(x)=x3−3x+1f(x) = x^3 - 3x + 1 is 3x2−33x^2 - 3.
Key Concept
Differentiation of Polynomial Functions
Explanation
The first derivative of a polynomial function is found by applying the power rule to each term separately. The power rule states that the derivative of xnx^n is nxn−1nx^{n-1}. Constants differentiate to zero.
What is the second derivative of the function f(x)?
Solution by Steps
step 1
To find the second derivative of the given function at x=0 x = 0 , we need to differentiate the series term by term twice and then evaluate at x=0 x = 0
step 2
The general term of the series is ln⁡(n)3nn3xn \frac{\ln(n)}{3^n n^3} x^n . The first derivative of this term with respect to x x is ln⁡(n)3nn3nxn−1 \frac{\ln(n)}{3^n n^3} n x^{n-1}
step 3
The second derivative of the general term with respect to x x is ln⁡(n)3nn3n(n−1)xn−2 \frac{\ln(n)}{3^n n^3} n(n-1) x^{n-2}
step 4
Evaluating the second derivative at x=0 x = 0 for n=2 n = 2 gives ln⁡(2)32232(2−1)⋅02−2 \frac{\ln(2)}{3^2 2^3} 2(2-1) \cdot 0^{2-2}
step 5
Since 00 0^{0} is undefined in general, but for the purposes of series expansion, it is considered to be 1 1 . Thus, the term simplifies to ln⁡(2)32232 \frac{\ln(2)}{3^2 2^3} 2
step 6
Simplifying the expression gives 2ln⁡(2)3223=ln⁡(2)3222 \frac{2 \ln(2)}{3^2 2^3} = \frac{\ln(2)}{3^2 2^2}
Answer
ln⁡(2)3222 \frac{\ln(2)}{3^2 2^2}
Key Concept
Term-by-term differentiation of power series
Explanation
The second derivative of a power series can be found by differentiating each term of the series twice with respect to x x , and then evaluating at a specific point, in this case, x=0 x = 0 .
What is the formula for finding the second derivative of a function f
Solution by Steps
step 1
To find the second derivative of a function f f , we denote it as d2dx2(f) \frac{d^2}{dx^2}(f)
step 2
According to the asksia-ll calculator, the second derivative of f f is d2df2(f)=0 \frac{d^2}{df^2}(f) = 0 . This implies that the second derivative is a constant value of zero
Answer
The second derivative of the function f f is 0.
Key Concept
Second Derivative of a Function
Explanation
The second derivative of a function measures the rate at which the first derivative changes, indicating the concavity of the function. If the second derivative is zero, it suggests that the function has a constant slope, meaning it is linear.
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