Let f(x) = | 3 sin x + 4 cos x | ((x + k)2 – pπ(x + k) + qπ2), where p, q, k \in R and k

Engineering

Mathematics

Important Points on Derivability

Question

Let f(x) = | 3 sin x + 4 cos x | ((x + k)² – pπ(x + k) + qπ²), where p, q, k $\in$ R and k > 0 is differentiable in (0, 2π) then

p + q = 7

p + q = 5

k = cos^–1 $(\frac{3}{5})$

k = sin^–1 $(\frac{3}{5})$

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f (x) = |3 sin x + 4 cos x| ((x + k)² – pπ(x + k) + qπ²)
= |5 sin (x + α)| ((x + k)² – pπ(x + k) + qπ²)
Let f(x) = | 5 sin (x + α) | (x² + bx + c)
sin (x + α) = 0 ⇒ x + α = π , 2π
   x = π – α, 2π – α where α = tan^–1
   For f(x) to be derivable in (0, 2π)
   quadratic equation x² + bx + c = 0 must have its roots as π – α and 2π – α.
$∴$ (x – π + α) (x – 2π + α) = 0
⇒ (x + α)² – 3π (x + α) + 2π² = 0
$∴$ k = α = tan^–1 $(\frac{4}{3})$ and p = 3, q = 2