Question

Verify the Rolle’s Theorem for the function f(x) = e^x cos x in `[- π/2, π/2]`

Answer 1

f(x) = e^x cos x in `[- π/2, π/2]`

`f(- π/2) = e^(- π/2) cos(- π/2)`

= `+ e^(- π/2) cos (π/2)`   ...[cos(– θ) = cos θ]

= `+ e^(- π/2) xx 0`   ...`[cos  π/2 = 0]`

= 0

`f(π/2) = e^(π/2) cos  π/2`

= `e^(π/2) xx 0`

= 0

Since Rolle’s theorem holds true, `f(- π/2) = f(π/2)`

Hence, there exists `c ∈ (- π/2, π/2)` such that f’(c) = 0

–e^c sin c + cos c · e^c = 0

sin c · e^c = e^c cos c

tan c = 1

`tan c = tan  π/4`

`c = π/4`

`c = π/4 ∈ [- π/2, π/2]`

Hence, Rolle’s theorem is verified.