A concave mirror for face viewing has a focal length of 0.4 m. The distance at which you hold the mirror from your face in order to see your image upright with a magnification of 5 is:

We have a concave mirror whose focal length is given as $$f = 0.4\ \text{m}.$$ According to the sign convention for spherical mirrors, the focal length of a concave mirror is taken as positive, so we shall keep $$f = +0.4\ \text{m}.$$

We want the image of the face to be upright and five times larger. For a mirror, the (linear) magnification formula is first stated:

$$m = -\dfrac{v}{u}$$

where $$u$$ is the object distance (measured from the pole, taken negative when the object is in front of the mirror) and $$v$$ is the image distance (taken positive for real images and negative for virtual images).

Because the image is specified to be upright, the magnification must be positive. We are told the magnification is $$m = +5,$$ so we write

$$+5 = -\dfrac{v}{u}.$$

Solving this for $$v$$ gives

$$v = -5u.$$

The negative value for $$v$$ (since $$u$$ is negative) confirms that the image is virtual and therefore forms behind the mirror, which is exactly what we expect for an upright image in a concave mirror.

Next, we invoke the mirror formula, which states:

$$\dfrac{1}{f} = \dfrac{1}{v} + \dfrac{1}{u}.$$

Substituting the known focal length $$f = 0.4\ \text{m}$$ and the expression $$v = -5u,$$ we get

$$\dfrac{1}{0.4} = \dfrac{1}{-5u} + \dfrac{1}{u}.$$

Evaluating the left-hand side,

$$\dfrac{1}{0.4} = 2.5.$$

So the equation becomes

$$2.5 = \dfrac{1}{-5u} + \dfrac{1}{u}.$$

We combine the two fractions on the right. Writing them with the common denominator $$5u,$$ we have

$$\dfrac{1}{-5u} + \dfrac{1}{u} = -\dfrac{1}{5u} + \dfrac{5}{5u} = \dfrac{4}{5u}.$$

Hence the equation is

$$2.5 = \dfrac{4}{5u}.$$

To isolate $$u,$$ we cross-multiply:

$$2.5 \times 5u = 4.$$

That gives

$$12.5u = 4.$$

Now dividing both sides by $$12.5,$$ we obtain

$$u = \dfrac{4}{12.5} = 0.32\ \text{m}.$$

Because the object distance in the sign convention is actually negative (the face is in front of the mirror), the formal value is $$u = -0.32\ \text{m}.$$ However, the distance you physically hold the mirror from your face is the magnitude of this quantity:

$$|u| = 0.32\ \text{m}.$$

Hence, the correct answer is Option C.