# If f(x) is continuous and differentiable and #f(x) = ax^4 + 5x#; #x<=2# and #bx^2 - 3x#; x> 2, then how do you find b?

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

To find the value of (b), we need to ensure that the function (f(x)) is continuous at (x = 2). For a function to be continuous at a point, the left-hand limit and the right-hand limit must be equal, and they must equal the value of the function at that point.

Given (f(x) = ax^4 + 5x) for (x \leq 2) and (f(x) = bx^2 - 3x) for (x > 2), we can set up the continuity condition:

[f(2) = \lim_{x \to 2^-} f(x) = \lim_{x \to 2^+} f(x)]

First, let's find (f(2)): [f(2) = a(2)^4 + 5(2) = 16a + 10]

Now, let's find the left-hand limit as (x) approaches (2): [\lim_{x \to 2^-} f(x) = \lim_{x \to 2^-} (ax^4 + 5x) = a(2)^4 + 5(2) = 16a + 10]

And the right-hand limit as (x) approaches (2): [\lim_{x \to 2^+} f(x) = \lim_{x \to 2^+} (bx^2 - 3x) = b(2)^2 - 3(2) = 4b - 6]

For (f(x)) to be continuous at (x = 2), the left-hand limit must equal the right-hand limit, so:

[16a + 10 = 4b - 6]

We can then solve this equation for (b): [4b = 16a + 10 + 6] [4b = 16a + 16] [b = 4a + 4]

So, the value of (b) is (4a + 4).

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 equation of the tangent line to the curve #y= (x-3) / (x-4)# at (5,2)?
- What is the equation of the line tangent to # f(x)=2/(4 − x^2)# at # x=3#?
- What is the slope of the line tangent to the graph of #x^2xy+y^2=7#?
- How do you find f'(x) using the limit definition given # f(x)= 2x^2-x#?
- What is the equation of the tangent line of #f(x)=x^3+2x^2-3x+2# at #x=1#?

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