Let be such that the function f ( x ) = { 2 ( x 2 - 2 ) + 2 x , x 1 + 3 ) x , x 1 be differentiable at all x. Then 34 is equal to [2026]

Question

Let $α, β \in ℝ$ be such that the function $f (x)$ $=$ ${\begin{matrix} 2 α (x^{2} - 2) + 2 β x, & x < 1 \\ (α + 3) x + (α - β), & x \geq 1 \end{matrix}$

be differentiable at all $x \in ℝ$ . Then $34 (α + β)$ is equal to [2026]

Smart Achievers · Accepted Answer

(4)

$f (x)$ $f (x)$ ${\begin{matrix} 2 α x^{2} + 2 β x - 4 α, & x < 1 \\ (α + 3) x + α - β, & x \geq 1 \end{matrix}$

$f (1^{+}) = 2 α - β + 3, f (1^{-}) = - 2 α + 2 β$

$2 α - β + 3 = 2 β - 2 α \Rightarrow 4 α - 3 β + 3 = 0 . . . (1)$

$f' (1^{+}) = 4 α + 2 β, f' (1^{-}) = α + 3$

$4 α + 2 β = α + 3 \Rightarrow 3 α + 2 β - 3 = 0 . . . (2)$

$Solving (1) & (2)$

We get $α = \frac{3}{17}, β = \frac{21}{17}$

$\Rightarrow 34 (α + β) = 34 \times \frac{27}{17} = 48$

Solution

Q.
Let $α, β \in ℝ$ be such that the function $f (x)$ $=$ ${\begin{matrix} 2 α (x^{2} - 2) + 2 β x, & x < 1 \\ (α + 3) x + (α - β), & x \geq 1 \end{matrix}$

be differentiable at all $x \in ℝ$ . Then $34 (α + β)$ is equal to [2026]

1 84

2 24

3 36

4 48

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