Question

question 7 (1 point)
the mean value theorem states that if a function f(x) is continuous on a,b and differentiable on (a,b), then there exists a point c in (a,b) such that the slope of the tangent line at c equals:
zero
the slope of the secant line between a and b
the average rate of change of f between 0 and c
f(c)

Explanation:

The Mean Value Theorem (MVT) formula is \(f^{\prime}(c)=\frac{f(b)-f(a)}{b - a}\). The right - hand side \(\frac{f(b)-f(a)}{b - a}\) is the slope of the secant line connecting the points \((a,f(a))\) and \((b,f(b))\).

• The slope of the tangent line at \(c\) is \(f^{\prime}(c)\).
• The slope of the secant line between \(a\) and \(b\) is \(\frac{f(b)-f(a)}{b - a}\), which is equal to \(f^{\prime}(c)\) by the MVT.
• The slope of the tangent line at \(c\) is not necessarily zero (that would be Rolle's Theorem when \(f(a)=f(b)\)).
• The average rate of change between \(0\) and \(c\) is \(\frac{f(c)-f(0)}{c-0}\), which is not related to the MVT formula.
• \(f(c)\) is the value of the function at \(c\), not a slope.

Answer:

the slope of the secant line between a and b