Find the constant a such that the function is continuous on the entire real numb

Ramsey

Ramsey

Answered question

2021-09-16

Find the constant a such that the function is continuous on the entire real number line.
f(x)={x3x2ax2x>2

Answer & Explanation

Neelam Wainwright

Neelam Wainwright

Skilled2021-09-17Added 102 answers

f(x)={x3x2ax2x>2 
Given that f is continuous down the complete real number line, x=2 is also where f is continuous.
This means
limx2+f(x)=limx2f(x)=limx2f(x)=f(2)
limx2+f(x)=ax2=4a
limx2f(x)=x3=23=8
limx2+f(x)=limx2f(x)=8
4a=8a=2
Result:
a=2.

Eliza Beth13

Eliza Beth13

Skilled2023-05-14Added 130 answers

Result:
The constant a can be any real number.
Solution:
To find the constant a such that the function f(x) is continuous on the entire real number line, we need to ensure that the two cases of the function match up at the point x=2.
Let's analyze the left-hand side of the function, x3, for x2. Since x3 is a polynomial, it is continuous for all real values of x.
Now, let's consider the right-hand side of the function, ax2, for x>2. To ensure continuity at x=2, the limit of ax2 as x approaches 2 from the left should be equal to the value of ax2 at x=2.
To express this mathematically, we can write:
limx2ax2=ax2|x=2
Evaluating the limit gives:
limx2ax2=a(2)2=4a
And the value of ax2 at x=2 is:
ax2|x=2=a(2)2=4a
Therefore, we need to find the value of a that satisfies the equation:
limx2ax2=ax2|x=2
Substituting the expressions we derived earlier, we have:
4a=4a
In conclusion, the constant a can be any real number.
madeleinejames20

madeleinejames20

Skilled2023-05-14Added 165 answers

Step 1:
Let's evaluate f(x) at x=2 in both cases:
For x2, we have:
f(x)=x3
Substituting x=2, we get:
f(2)=23=8
For x>2, we have:
f(x)=ax2
Substituting x=2, we get:
f(2)=a(2)2=4a
Step 2:
To ensure continuity at x=2, the two expressions for f(2) must be equal. Therefore, we have:
8=4a
Solving this equation for a, we find:
a=84=2
Hence, the constant a that makes the function f(x) continuous on the entire real number line is a=2.
nick1337

nick1337

Expert2023-05-14Added 777 answers

To find the constant a such that the function f(x) is continuous on the entire real number line, we need to ensure that the left-hand limit of f(x) as x approaches 2 is equal to the right-hand limit of f(x) at x=2, and they are both equal to the value of f(x) at x=2.
Let's calculate these limits and evaluate the function at x=2 to find the value of a.
The left-hand limit of f(x) as x approaches 2 is given by:
limx2f(x)=limx2x3
Since x3 is a continuous function, we can directly substitute x=2 into the function:
limx2f(x)=23=8
The right-hand limit of f(x) as x approaches 2 is given by:
limx2+f(x)=limx2+ax2
Again, we can substitute x=2 into the function:
limx2+f(x)=a·22=4a
To ensure continuity, we set these two limits equal to each other:
8=4a
Now, we can solve for a by dividing both sides of the equation by 4:
84=4a4
Simplifying, we have:
2=a
Hence, the constant a that makes the function f(x) continuous on the entire real number line is 2.

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