Question

10. Find the values of k for which the function f(x) = −4x 2 + (4k − 1)x − k 2 + 4 is negative for all values of x.

Answer 1

The function $f(x)=-4x^2+(4k-1)x-k^2+4$ is a parabola. Therefore, $f$ does not intersect the abscissa axis, i.e. $-4x^2+(4k-1)x-k^2+4=0$ has no solution.

Therefore, the discriminant must be negative:

$$\Delta =b^2-4ac<0\Rightarrow (4k-1)^2-4\cdot (-4)\cdot (-k^2+4)<0$$

$$\Rightarrow (4k)^2-2\cdot 4k\cdot 1+1^2+16(-k^2+4)<0$$

$$\Rightarrow 16k^2-8k+1-16k^2+64<0\Rightarrow 65<8k$$

$$\Rightarrow k>\dfrac{65}{8}$$

Therefore, the values of $k$ that satisfy the condition are those that satisfy that $k>\dfrac{65}{8}$ .