If the function f(x)=⎩⎨⎧x2{sin(k1+1)x+sin(k2−1)x},4,x2loge(2+k2x2+k1x),x<0x=0x>0 is continuous at x=0, then k12+k22 is
Get the complete, step-by-step math solution for: "If the function f(x) = (2)/(x) \{ sin (k_1 + 1) x + sin (k_2 - 1) x \}, & x < 0 \\ 4, & x = 0 \\ (2)/(x) _e (2 + k_1 x)/(2 + k_2 x) , & x > 0 is conti...". Powered by SolveForX AI math tutor.
Step-by-Step Solution
Step 1: Apply Continuity Condition
For a function to be continuous at a point x=0, the left-hand limit, the right-hand limit, and the function's value at that point must all be equal. We are given f(0)=4. Therefore, we need to find the left-hand limit and the right-hand limit and set them equal to 4.
Step 2: Evaluate Left-Hand Limit
We evaluate the left-hand limit using the standard limit limx→0xsinax=a. Applying this, we get 2((k1+1)+(k2−1)), which simplifies to 2(k1+k2). Since the function is continuous, this limit must be equal to f(0)=4.
Step 3: Set Left-Hand Limit Equal to f(0)
From the previous step, we found the left-hand limit to be 2(k1+k2). Equating this to f(0)=4, we get the equation 2(k1+k2)=4, which simplifies to k1+k2=2. This is our first equation relating k1 and k2.
Step 4: Evaluate Right-Hand Limit
We evaluate the right-hand limit. We use the property of logarithms loge(A/B)=logeA−logeB. Then, we apply the standard limit limx→0xloge(1+ax)=a. To use this, we rewrite loge(2+kx) as loge(2(1+kx/2))=loge2+loge(1+kx/2). The loge2 terms cancel out, leaving us with 2(2k1−2k2), which simplifies to k1−k2.
Step 5: Set Right-Hand Limit Equal to f(0)
Since the function is continuous, the right-hand limit must also be equal to f(0)=4. Therefore, we have the second equation: k1−k2=4.
Step 6: Solve System of Equations
Now we have a system of two linear equations: k1+k2=2 and k1−k2=4. Adding these two equations eliminates k2, giving 2k1=6, so k1=3. Substituting k1=3 into the first equation gives 3+k2=2, which means k2=−1.
Step 7: Calculate k12+k22
Finally, we substitute the values of k1=3 and k2=−1 into the expression k12+k22. This gives 32+(−1)2=9+1=10.