If $f(x) = \begin{cases} ax + b, & x \leq 3 \\ 7, & 5<x \end{cases}$ is continuous in $\mathbb{R}$, then the values of $a$ and $b$ are:

Question

Let $[t]$ denote the greatest integer less than or equal to $t$.
If the function
\[ f(x)= \begin{cases} b^2 \sin\!\left[\dfrac{\pi}{2}\left[\dfrac{\pi}{2}(\cos x+\sin x)\cos x\right]\right], & x < 0 \\ \dfrac{\sin x-\dfrac{1}{2}\sin 2x}{x^3}, & x > 0 \\ a, & x = 0 \end{cases} \] is continuous at $x=0$, then $a^2+b^2$ is equal to

Answer 1

For continuity at $x = 3$ and $x = 5$, left-hand and right-hand limits must be equal. At $x = 3$, the condition is: \[ a(3) + b = 7 \quad \text{(matching the second function part)} \] This yields $3a + b = 7$. For the condition at $x = 5$: \[ a(5) + b = 7 \quad \text{(again, matching the second function part)} \] Solving these simultaneous equations results in $a = -3$ and $b = -8$.

If $f(x) = \begin{cases} ax + b, & x \leq 3 \\ 7, & 5<x \end{cases}$ is continuous in $\mathbb{R}$, then the values of $a$ and $b$ are:

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The Correct Option is C

Solution and Explanation

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