新增关于幂指对函数的一些自拟题目（严捷）

题库0.3/Problems.json

@@ -505373,6 +505373,126 @@
         "space": "",
         "unrelated": []
     },
+    "019800": {
+        "id": "019800",
+        "content": "已知幂函数 $f(x)=x^m$.\\\\\n    (1) 若 $m>0$, 求证: $f(x)$ 在 $[0,+\\infty)$ 上为严格增函数;\\\\\n    (2) 若 $m<0$, 求证: $f(x)$ 在 ($0,+\\infty$) 上为严格减函数.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "自拟题目",
+        "edit": [
+            "20230802\t严捷"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "019801": {
+        "id": "019801",
+        "content": "证明指数函数 $y=a^x$ 与 $y=(\\dfrac{1}{a})^x$($a>0$ 且 $a \\neq 1$) 的图像关于 $y$ 轴对称.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "自拟题目",
+        "edit": [
+            "20230802\t严捷"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "019802": {
+        "id": "019802",
+        "content": "已知 $f(x)=3^x$, $u, v \\in \\mathbf{R}$.\\\\\n    (1) 求证: 对于任意 $u, v$ 都有 $f(u) \\cdot f(v)=f(u+v)$;\\\\\n    (2) 写出一个关于 $\\dfrac{f(u)}{f(v)}$ 的类似等式.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "自拟题目",
+        "edit": [
+            "20230802\t严捷"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "019803": {
+        "id": "019803",
+        "content": "证明: 当 $a>1$, $N>1$ 时, $\\log _a N>0$.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "自拟题目",
+        "edit": [
+            "20230802\t严捷"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "019804": {
+        "id": "019804",
+        "content": "证明: 当 $a>1$ 时, 对数函数 $y=\\log _a x$ 在区间 ($0,+\\infty$) 上是严格增函数.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "自拟题目",
+        "edit": [
+            "20230802\t严捷"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "019805": {
+        "id": "019805",
+        "content": "已知 $f(x)=\\log _a(2-a x)$.\\\\\n    (1) 若 $f(x)$ 自然意义下的定义域是 ($-\\infty, 3$), 求 $a$ 的值并指出 $f(x)$ 的单调性;\\\\\n    (2) 若 $f(x)$ 在区间 $[0,1]$ 上是严格减函数, 求 $a$ 的取值范围.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "自拟题目",
+        "edit": [
+            "20230802\t严捷"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
     "020001": {
         "id": "020001",
         "content": "判断下列各组对象能否组成集合, 若能组成集合, 指出是有限集还是无限集.\\\\\n(1) 上海市控江中学$2022$年入学的全体高一年级新生;\\\\\n(2) 中国现有各省的名称;\\\\\n(3) 太阳、$2$、上海市;\\\\\n(4) 大于$10$且小于$15$的有理数;\\\\\n(5) 末位是$3$的自然数;\\\\\n(6) 影响力比较大的中国数学家;\\\\\n(7) 方程$x^2+x-3=0$的所有实数解;\\\\ \n(8) 函数$y=\\dfrac 1x$图像上所有的点;\\\\ \n(9) 在平面直角坐标系中, 到定点$(0, 0)$的距离等于$1$的所有点;\\\\\n(10) 不等式$3x-10<0$的所有正整数解;\\\\\n(11) 所有的平面四边形.",