- Difficulty
- Standard
- Suggested time
- 3 min
- Marks
- 2
- Theory
- CLP-1 §2.13Key results
The Mean Value Theorem practice question 7
A function satisfies for every real , and . Find .
You know the slope everywhere and one value of . Which theorem links the change to a value of ?
Apply the Mean Value Theorem on : for some .
You do not need to know : whatever it is, . Rearrange for .
A whole number.
Theory CLP-1 §2.13Key results
Worked solution
Strategy
A constant derivative forces a constant secant slope, by the MVT. That pins down the value at 5 with no formula for f needed; the consequence is that f must be the line through (1, 2) with slope 3.
Working
f is differentiable everywhere, so it is continuous on [1, 5] and the MVT applies there; whatever c is, f′(c) = 3.
Rearrange.
Answer
The answer is 14.
Check
The line 3x − 1 fits both facts, slope 3 and value 2 at x = 1, and gives 14 at x = 5.
Where this goes wrong
Answering 15 by computing 3 × 5, as if f(x) = 3x. The slope is 3, but the line passes through (1, 2), not the origin: f(x) = 3x − 1.
Leans on
Slope of a line, from Functions, graphs and inverses
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