1. Verify mean value theorem for the function f(x) = (x – 2)^2 in [1, 4].

Question

1. Verify mean value theorem for the function f(x) = (x – 2)² in [1, 4].
2. Find a point on the curve y = (x – 2)² at which the tangent is parallel to the chord joining the points (1, 1) and (4, 2)
3. Find a point on the above curve at which the tangent is parallel to the x-axis.

Answer 1

1. f(x) = (x – 1)², x ∈ [1, 4]
f(x) is continuous in [1, 4]
f'(x) = 2(x – 2) is differentiable in [1, 4]
Then there exists c ∈ [1, 4] so that

Hence Mean Value Theorem is verified.

2. c = \(\frac{5}{2}\) will be the x-coordinate to the point of contact of tangent and the curve, then 

y = (x – 2)² 

⇒ y = \((\frac{5}{2} -2)^2\) = \(\frac{1}{4}\)

Therefore the point is (\(\frac{5}{2}\),\(\frac{1}{4}\))

3. The tangent parallel to x- axis will have
f'(c) = 0 ⇒ 2(c – 2) = 0 ⇒ c = 2
Then; x = 2 ⇒ y = (2 – 2)² = 0
Therefore the point is (2, 0).