Question

Suppose that \(f\) has a negative derivative for all values of \(x\) and that \(f(1)=0 .\) Which of the following statements must be true of the function\(h (x)=\int_{0}^{x} f(t) d t ?\) Give reasons for your answers. (a) \(h\) is a twice-different table function of \(x\) . (b) \(h\) and \(d h / d x\) are both continuous (c) The graph of \(h\) has a horizontal tangent at \(x=1\) (d) \(h\) has a local maximum at \(x=1\) (e) \(h\) has a local minimum at \(x=1\) (f) The graph of \(h\) has an inflection point at \(x=1\) (g) The graph of \(d h / d x\) crosses the \(x\) -axis at \(x=1\)

Short answer

(a) True, (b) True, (c) True, (d) True, (e) False, (f) False, (g) True

Step-by-step solution

Twice Differentiability

By the second fundamental theorem of calculus, h(x) = \(\int_{0}^{x} f(t) d t\) which implies that \(h'(x)=f(x)\). Given that \(f\) has a negative derivative (i.e., \(f'(x)\) exists), this further implies \(h''(x)=f'(x)\) exists. Hence, h(x) is twice differentiable. So, (a) is True.

Continuity analysis

As per the properties of integration and differentiation, both \(h(x)\) and \(\frac{dh}{dx}\) are continuous functions. Hence, (b) is True.

Examination of the Tangent at \(x=1\)

Given \(h'(x) = f(x)\), when \(x=1\), \(h'(1) = f(1) = 0\). Therefore, the graph of h(x) has a horizontal tangent at \(x=1\), and (c) is True.

Analyzing Local Extrema

With \(h'(x) = f(x)\), and given that \(f(x)\) has a negative derivative for all \(x\), it means \(f(x)\) is decreasing everywhere. This means that \(h'(x)\) is also decreasing everywhere. So, when \(h'(x)\) transitions from positive to negative at \(x=1\), it signifies a local maximum. Thus, (d) is True.

- Local minimum and Inflection Point Analysis

(e) and (f) both are False because there's no evidence that \(h\) has a local minimum or an inflection point at \(x=1\). Given the above analysis, it's clear that \(h\) has a local maximum, not a minimum, at \(x=1\). Also, an inflection point would require a change in concavity, but there's no change in \(f\)'s sign to suggest that.

Cross of Derivate Graph at \(x=1\)

Given that \(h'(x) = f(x)\), and \(f(1) = 0\), the derivative of function \(h\), \(dh/dx = h'(x)\), does indeed cross the x-axis at \(x=1\). So, (g) is True.