Find the constant a such that the function is continuous on the entire real number line. f(x)={x3x≤2ax2x&gt;2

Question

Find the constant a such that the function is continuous on the entire real number line. f(x)={x3x≤2ax2x&amp;gt;2

Neelam Wainwright · Accepted Answer

f(x)={x3x≤2ax2x&amp;gt;2&amp;nbsp;Given that f is continuous down the complete real number line, x=2 is also where f is continuous.This meanslimx→2+f(x)=limx→2−f(x)=limx→2f(x)=f(2)∴limx→2+f(x)=ax2=4a∴limx→2−f(x)=x3=23=8∵limx→2+f(x)=limx→2−f(x)=8∴4a=8⇒a=2Result:a=2.

Eliza Beth13 · Accepted Answer

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&#039;s analyze the left-hand side of the function, x3, for x≤2. Since x3 is a polynomial, it is continuous for all real values of x.Now, let&#039;s consider the right-hand side of the function, ax2, for x&amp;gt;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:limx→2−ax2=ax2|x=2Evaluating the limit gives:limx→2−ax2=a(2)2=4aAnd the value of ax2 at x=2 is:ax2|x=2=a(2)2=4aTherefore, we need to find the value of a that satisfies the equation:limx→2−ax2=ax2|x=2Substituting the expressions we derived earlier, we have:4a=4aIn conclusion, the constant a can be any real number.

madeleinejames20 · Accepted Answer

Step 1:Let&#039;s evaluate f(x) at x=2 in both cases:For x≤2, we have:f(x)=x3Substituting x=2, we get:f(2)=23=8For x&amp;gt;2, we have:f(x)=ax2Substituting x=2, we get:f(2)=a(2)2=4aStep 2:To ensure continuity at x=2, the two expressions for f(2) must be equal. Therefore, we have:8=4aSolving this equation for a, we find:a=84=2Hence, the constant a that makes the function f(x) continuous on the entire real number line is a=2.

nick1337 · Accepted Answer

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&#039;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:limx→2−f(x)=limx→2−x3Since x3 is a continuous function, we can directly substitute x=2 into the function:limx→2−f(x)=23=8The right-hand limit of f(x) as x approaches 2 is given by:limx→2+f(x)=limx→2+ax2Again, we can substitute x=2 into the function:limx→2+f(x)=a·22=4aTo ensure continuity, we set these two limits equal to each other:8=4aNow, we can solve for a by dividing both sides of the equation by 4:84=4a4Simplifying, we have:2=aHence, the constant a that makes the function f(x) continuous on the entire real number line is 2.

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

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