# We have #f:RR->RR,f(x)=|x|root(3)(1-x^2)#.How to find maximum domain of differentiability?

See below.

So,

The function's graph is finally shown here: graph{y=(1-x^2)^(1/3)absx [-2.433, 2.436, -1.215, 1.217]}

By signing up, you agree to our Terms of Service and Privacy Policy

To find the maximum domain of differentiability for the function ( f(x) = |x|\sqrt[3]{1 - x^2} ), we need to consider the points where the function might not be differentiable. Since the function involves a square root, it might not be differentiable at points where the expression under the square root becomes negative.

First, we need to identify the critical points where the function might not be differentiable. These occur when the expression ( 1 - x^2 ) under the square root becomes negative, as the square root of a negative number is not defined in the real number system.

So, we solve the inequality ( 1 - x^2 \geq 0 ) to find the critical points.

( 1 - x^2 \geq 0 )

( 1 \geq x^2 )

( x^2 \leq 1 )

( -1 \leq x \leq 1 )

Therefore, the critical points occur when ( -1 \leq x \leq 1 ).

However, since the function involves the absolute value of ( x ), the function is actually symmetrical about the y-axis. So, any critical point within the domain will have a corresponding critical point on the other side of the y-axis. Thus, the critical points actually occur when ( -1 \leq x \leq 1 ) and ( -1 \geq x \geq -1 ), which simplifies to ( -1 \leq x \leq 1 ).

Hence, the maximum domain of differentiability for the function ( f(x) = |x|\sqrt[3]{1 - x^2} ) is ( -1 \leq x \leq 1 ).

By signing up, you agree to our Terms of Service and Privacy Policy

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- How do you find the second derivative of #x^2y^2#?
- How do you differentiate #f(x)=(lnx)(ln3x) # using the product rule?
- How do you find the derivative of #f(x)= e^x((x^3)-1)#?
- How do you differentiate #f(x)= (1 - sin^2x)/(1 - sinx)^2 # using the quotient rule?
- How do you use the Product Rule to find the derivative of #(7x^4 + 2x^6) sin(7x)#?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7