If the function `f(x)=2x^2-k x+5` is increasing on `[1,\ 2]` , then `k` lies in the interval (a) `(-oo,\ 4)` (b) `(4,\ oo)` (c) `(-oo,\ 8)` (d) `(8,\ oo)`

Options

A. `(-oo,4)`
B. `(4,oo)`
C. `(-oo,8)`
D. `(8,oo)`

Answer

A

Explanation

We have f'(x) = 4x-k
Since f(x) is an increasing function [1,2] Therefore,
f'(x) gt for 1 `le` x `le` 2
Now ,
f"(x) = 4 for all x `in` [1,2]
`rArr f''(x) gt0 " for all " x in [1,2]`
` rArr` f'(x) is an increasing function on [1,2]
`rArr` f'(1) is the least value of f'(x) on [1,2]
therefore for f'(x) to be greater than zero for all x in [1,2] its least value i.e. f'(1) must also be greater than zero .
Now, `f'(1) gt0rArr4-krArr0rArrklt4rArrk in (-oo,4).`